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Algebra Formulas

85 formulas
Arithmetic SequencesGo to
Geometric SequencesGo to
Harmonic SequencesGo to
Recursive SequencesGo to
Figurate NumbersGo to
EquationsGo to
Logarithm RulesGo to
Identities & FactoringGo to
Exponent RulesGo to
Radical RulesGo to
Polynomial TheoremsGo to

Arithmetic Sequences

(6 formulas)

Common Difference

an+1−an=da_{n+1} - a_n = d
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The defining property of an arithmetic sequence: the difference between any two consecutive terms is the same constant dd. The value of dd determines whether the sequence increases (d>0d > 0), decreases (d<0d < 0), or stays constant (d=0d = 0).
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General Term (Arithmetic Sequence)

an=a1+(n−1)da_n = a_1 + (n - 1)d
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Explicit (closed-form) expression for the nn-th term of an arithmetic sequence: start at a1a_1 and add the common difference dd exactly n−1n - 1 times. The formula is linear in nn — plotted against the index, the terms lie on a straight line with slope dd.
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Recursive Form (Arithmetic Sequence)

a1=c,an=an−1+da_1 = c, \quad a_n = a_{n-1} + d
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Recursive definition of an arithmetic sequence: each term is the previous term plus the common difference dd, with initial value a1=ca_1 = c. Applying the rule n−1n - 1 times recovers the explicit form an=c+(n−1)da_n = c + (n-1)d.
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Arithmetic Series Sum

Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)
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explanationderivationvariantsrelated formulasrelated definitions
Closed-form sum of the first nn terms of an arithmetic sequence. Famously the formula behind the Gauss schoolchild story: pairing the first and last terms gives a1+ana_1 + a_n, and there are n/2n/2 such pairs.
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Arithmetic Mean

M=a+b2M = \frac{a + b}{2}
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The arithmetic mean (average) of two numbers aa and bb is their sum divided by 22. Generalizes to nn values as the sum divided by nn.
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Arithmetic Mean Property

an=an−1+an+12a_n = \frac{a_{n-1} + a_{n+1}}{2}
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In an arithmetic sequence, every interior term is the arithmetic mean of its two neighbors. This follows directly from the constant-difference property: an−an−1=an+1−ana_n - a_{n-1} = a_{n+1} - a_n implies an−1+an+1=2ana_{n-1} + a_{n+1} = 2a_n.
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Geometric Sequences

(7 formulas)

Common Ratio

an+1an=r\frac{a_{n+1}}{a_n} = r
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The defining property of a geometric sequence: the ratio of any two consecutive terms is the same constant rr. The sign and magnitude of rr determine growth, decay, oscillation, or convergence.
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General Term (Geometric Sequence)

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
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Explicit expression for the nn-th term of a geometric sequence: start at a1a_1 and multiply by the common ratio rr exactly n−1n - 1 times. The formula is exponential in nn — terms scale by a constant factor between consecutive indices.
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Recursive Form (Geometric Sequence)

a1=c,an=r⋅an−1a_1 = c, \quad a_n = r \cdot a_{n-1}
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Recursive definition of a geometric sequence: each term is the previous term multiplied by the common ratio rr, with initial value a1=ca_1 = c. Applying the rule n−1n - 1 times gives the explicit form an=c⋅rn−1a_n = c \cdot r^{n-1}.
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Finite Geometric Series Sum

Sn=a1⋅1−rn1−rS_n = a_1 \cdot \frac{1 - r^n}{1 - r}
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Closed-form sum of the first nn terms of a geometric sequence with common ratio r≠1r \neq 1. Derived by multiplying the sum by rr, subtracting, and solving.
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Infinite Geometric Series Sum

S=a11−rS = \frac{a_1}{1 - r}
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When ∣r∣<1|r| < 1, the partial sums SnS_n converge as n→∞n \to \infty. Since rn→0r^n \to 0, the finite formula collapses to a1/(1−r)a_1 / (1 - r). This is the rare case where an infinite sum has a clean closed form.
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Geometric Mean

G=abG = \sqrt{ab}
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The geometric mean of two positive numbers aa and bb is the square root of their product. Generalizes to nn positive values as the nn-th root of their product.
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Geometric Mean Property

an=an−1⋅an+1a_n = \sqrt{a_{n-1} \cdot a_{n+1}}
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In a geometric sequence with positive terms, every interior term is the geometric mean of its two neighbors. This follows from an−1=an/ra_{n-1} = a_n / r and an+1=anra_{n+1} = a_n r, so an−1⋅an+1=an2a_{n-1} \cdot a_{n+1} = a_n^2.
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Harmonic Sequences

(3 formulas)

General Term (Harmonic Sequence)

an=1b1+(n−1)da_n = \frac{1}{b_1 + (n-1)d}
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The nn-th term of a harmonic sequence is the reciprocal of the nn-th term of an arithmetic sequence with first term b1b_1 and common difference dd. The simplest case takes bn=nb_n = n, giving the natural-number reciprocals 1,12,13,…1, \frac{1}{2}, \frac{1}{3}, \ldots.
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Harmonic Mean

H=n1a1+1a2+⋯+1anH = \frac{n}{\dfrac{1}{a_1} + \dfrac{1}{a_2} + \cdots + \dfrac{1}{a_n}}
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The harmonic mean of nn positive numbers is nn divided by the sum of their reciprocals. Equivalently, it is the reciprocal of the arithmetic mean of the reciprocals. Used for averaging rates.
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AM-GM-HM Inequality

H≤G≤AH \leq G \leq A
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For any set of positive numbers, the harmonic mean is at most the geometric mean, which is at most the arithmetic mean. Equality holds throughout if and only if all values are identical.
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Recursive Sequences

(11 formulas)

Fibonacci Recurrence

F1=1,F2=1,Fn=Fn−1+Fn−2F_1 = 1, \quad F_2 = 1, \quad F_n = F_{n-1} + F_{n-2}
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Defines the Fibonacci sequence by a two-term linear recurrence with initial values F1=F2=1F_1 = F_2 = 1. Each subsequent term is the sum of its two immediate predecessors, producing 1,1,2,3,5,8,13,21,…1, 1, 2, 3, 5, 8, 13, 21, \ldots.
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Golden Ratio

ϕ=1+52\phi = \frac{1 + \sqrt{5}}{2}
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The golden ratio ϕ≈1.618\phi \approx 1.618 is the positive root of x2=x+1x^2 = x + 1. It equals the limit of consecutive Fibonacci ratios Fn+1/FnF_{n+1}/F_n and is the dominant growth rate of the Fibonacci sequence.
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Binet's Formula

Fn=ϕn−ψn5F_n = \frac{\phi^n - \psi^n}{\sqrt{5}}
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Closed-form expression for the nn-th Fibonacci number using the two roots of the characteristic equation x2=x+1x^2 = x + 1. Despite the irrationals ϕ,ψ,5\phi, \psi, \sqrt{5}, the result is always an integer — the irrational parts cancel exactly.
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Cassini's Identity

Fn−1Fn+1−Fn2=(−1)nF_{n-1} F_{n+1} - F_n^2 = (-1)^n
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The product of the two Fibonacci numbers flanking FnF_n, minus the square of FnF_n itself, alternates between +1+1 and −1-1 as nn changes parity. For n=6n = 6: F5F7−F62=5⋅13−64=1F_5 F_7 - F_6^2 = 5 \cdot 13 - 64 = 1.
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Sum of Fibonacci Numbers

∑k=1nFk=Fn+2−1\sum_{k=1}^{n} F_k = F_{n+2} - 1
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The sum of the first nn Fibonacci numbers is one less than a Fibonacci number two positions further along. Running totals always land just short of a future Fibonacci value.
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Sum of Squared Fibonacci Numbers

∑k=1nFk2=Fn⋅Fn+1\sum_{k=1}^{n} F_k^2 = F_n \cdot F_{n+1}
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The sum of the first nn squared Fibonacci numbers equals the product of FnF_n and Fn+1F_{n+1}. Has a geometric interpretation: stacking squares of side FkF_k tiles a rectangle of dimensions Fn×Fn+1F_n \times F_{n+1}.
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Fibonacci GCD Identity

gcd⁡(Fm,Fn)=Fgcd⁡(m,n)\gcd(F_m, F_n) = F_{\gcd(m, n)}
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The greatest common divisor of two Fibonacci numbers is itself a Fibonacci number, indexed by the GCD of the original indices. Connects the multiplicative structure of the Fibonacci sequence to the GCD of ordinary integers.
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Lucas Recurrence

L1=1,L2=3,Ln=Ln−1+Ln−2L_1 = 1, \quad L_2 = 3, \quad L_n = L_{n-1} + L_{n-2}
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The Lucas sequence uses the same recurrence as Fibonacci but starts from different initial values L1=1,L2=3L_1 = 1, L_2 = 3, producing 1,3,4,7,11,18,29,47,…1, 3, 4, 7, 11, 18, 29, 47, \ldots. Like Fibonacci, the ratio of consecutive Lucas numbers converges to the golden ratio.
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Lucas-Fibonacci Relation

Ln=Fn−1+Fn+1L_n = F_{n-1} + F_{n+1}
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Each Lucas number equals the sum of the two Fibonacci numbers flanking the same position. For n=5n = 5: L5=F4+F6=3+8=11L_5 = F_4 + F_6 = 3 + 8 = 11.
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Lucas Binet Formula

Ln=ϕn+ψnL_n = \phi^n + \psi^n
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Closed-form expression for the nn-th Lucas number using the same two roots ϕ,ψ\phi, \psi that appear in Binet's formula. Where Fibonacci subtracts the powers and divides by 5\sqrt{5}, Lucas adds them directly.
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Prime Number Theorem

π(n)≈nln⁡n\pi(n) \approx \frac{n}{\ln n}
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The number of primes up to nn, denoted π(n)\pi(n), is asymptotically n/ln⁡nn / \ln n. The ratio π(n)ln⁡n/n→1\pi(n) \ln n / n \to 1 as n→∞n \to \infty. Practically, near a large nn, roughly one in every ln⁡n\ln n integers is prime.
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Figurate Numbers

(10 formulas)

Triangular Number Formula

Tn=n(n+1)2T_n = \frac{n(n+1)}{2}
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Closed form for the nn-th triangular number — the sum of the first nn positive integers, or equivalently the number of dots in a triangular grid with nn rows. The first values are 1,3,6,10,15,21,28,36,…1, 3, 6, 10, 15, 21, 28, 36, \ldots.
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Recursive Form (Triangular Numbers)

T1=1,Tn=Tn−1+nT_1 = 1, \quad T_n = T_{n-1} + n
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Each triangular number is the previous one plus a new row of nn dots. The increments 1,2,3,4,…1, 2, 3, 4, \ldots form an arithmetic sequence with common difference 11.
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Triangular Number as Binomial Coefficient

Tn=(n+12)T_n = \binom{n+1}{2}
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The nn-th triangular number equals the number of ways to choose 22 items from n+1n + 1. For instance, T4=10T_4 = 10 equals the number of distinct handshakes among 55 people. Places triangular numbers inside combinatorics.
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Sum of Consecutive Triangular Numbers

Tn+Tn−1=n2T_n + T_{n-1} = n^2
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The sum of two consecutive triangular numbers is always a perfect square. Geometrically, two triangles of consecutive sizes fit together to form a square — cut a square grid of n2n^2 dots along its staircase diagonal to see why.
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Sum of Triangular Numbers

∑k=1nTk=n(n+1)(n+2)6\sum_{k=1}^{n} T_k = \frac{n(n+1)(n+2)}{6}
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The partial sums of the triangular numbers are the tetrahedral numbers — counts of dots arranged in successively larger tetrahedra. Extends the figurate-number construction from two dimensions to three.
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Square Number Formula

Sn=n2S_n = n^2
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The nn-th square number is the product of nn with itself, arrangeable as a square grid of n2n^2 dots. The first values are 1,4,9,16,25,36,49,64,81,100,…1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots.
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Recursive Form (Square Numbers)

S1=1,Sn=Sn−1+(2n−1)S_1 = 1, \quad S_n = S_{n-1} + (2n - 1)
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Each square number is built from the previous one by adding an L-shaped border — a gnomon — of 2n−12n - 1 dots. The increments 1,3,5,7,…1, 3, 5, 7, \ldots are the odd numbers.
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Sum of First n Odd Numbers

1+3+5+⋯+(2n−1)=n21 + 3 + 5 + \cdots + (2n - 1) = n^2
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The sum of the first nn odd numbers is exactly n2n^2. Follows from the arithmetic series formula with a1=1a_1 = 1 and an=2n−1a_n = 2n - 1: Sn=n2(1+2n−1)=n2S_n = \frac{n}{2}(1 + 2n - 1) = n^2.
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Sum of Squares

∑k=1nk2=n(n+1)(2n+1)6\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}
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Closed-form for the sum of the first nn perfect squares. The polynomial is cubic in nn — each new term contributes a squared value, so the running total grows faster than for triangular numbers.
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Pythagorean Triple Generator

a=m2−n2,b=2mn,c=m2+n2a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2
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Every primitive Pythagorean triple (a,b,c)(a, b, c) satisfying a2+b2=c2a^2 + b^2 = c^2 can be generated from a pair of integers m,nm, n. For m=2,n=1m = 2, n = 1: (3,4,5)(3, 4, 5). For m=3,n=2m = 3, n = 2: (5,12,13)(5, 12, 13). Non-primitive triples are integer multiples of primitive ones.
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Equations

(6 formulas)

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
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Gives the two solutions of any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 directly from its coefficients. The ±\pm produces two values: one using ++, the other using −-. The expression under the square root, b2−4acb^2 - 4ac, is the discriminant — it determines whether the roots are real and distinct, real and equal, or complex.
Function machine
a, b, c
√Δ
x₁, x₂
compute discriminant b² − 4ac
apply ± and divide by 2a
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Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac
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The discriminant of a quadratic ax2+bx+c=0ax^2 + bx + c = 0 determines the nature of its roots. If Δ>0\Delta > 0: two distinct real roots. If Δ=0\Delta = 0: one repeated real root. If Δ<0\Delta < 0: two complex conjugate roots.
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Square Root Property

x2=p  ⟹  x=±px^2 = p \implies x = \pm\sqrt{p}
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If a squared expression equals a constant, the variable equals the positive or negative square root of that constant. This is the simplest method for solving quadratics with no linear term.
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Completing the Square

x2+bx=(x+b2)2−b24x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \frac{b^2}{4}
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Rewrites a quadratic expression as a perfect square minus a constant. Take half the coefficient of xx, square it, add and subtract.
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Absolute Value Equation

∣p∣=b  ⟹  p=b or p=−b(b>0)|p| = b \implies p = b \text{ or } p = -b \quad (b > 0)
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An absolute value equation splits into two cases: the expression inside equals the positive value, or the expression inside equals the negative value.
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Absolute Value Inequalities

∣p∣<b  ⟹  −b<p<b∣p∣>b  ⟹  p<−b or p>b|p| < b \implies -b < p < b \qquad |p| > b \implies p < -b \text{ or } p > b
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A less-than inequality produces a compound inequality (a bounded interval). A greater-than inequality produces a disjunction (two unbounded rays).
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Logarithm Rules

(8 formulas)

Product Rule (Logarithms)

log⁡a(xy)=log⁡a(x)+log⁡a(y)\log_a(xy) = \log_a(x) + \log_a(y)
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The logarithm of a product equals the sum of the logarithms. This converts multiplication inside the argument into addition outside.
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Quotient Rule (Logarithms)

log⁡a(xy)=log⁡a(x)−log⁡a(y)\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)
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The logarithm of a quotient equals the difference of the logarithms. Division inside the argument becomes subtraction outside.
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Power Rule (Logarithms)

log⁡a(xn)=n⋅log⁡a(x)\log_a(x^n) = n \cdot \log_a(x)
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An exponent inside the argument moves out front as a multiplier. This is the key property that makes logarithms useful for simplifying expressions with exponents.
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Change of Base Formula

log⁡a(x)=log⁡b(x)log⁡b(a)\log_a(x) = \frac{\log_b(x)}{\log_b(a)}
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Converts a logarithm from one base to another. Most commonly used with b=10b = 10 or b=eb = e to evaluate logarithms on a calculator, which only has log⁡\log and ln⁡\ln keys.
Function machine
log_a(x)
log_b(x) and log_b(a)
log_b(x) / log_b(a)
compute log_b of both x and a
divide
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Logarithm of the Base

log⁡a(a)=1\log_a(a) = 1
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The logarithm of the base itself always equals 1, because a1=aa^1 = a.
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Logarithm of One

log⁡a(1)=0\log_a(1) = 0
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The logarithm of 1 is always 0, regardless of the base, because a0=1a^0 = 1.
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Logarithm of an Exponential

log⁡a(ax)=x\log_a(a^x) = x
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Applying a logarithm to its own base's exponential cancels both operations, returning the exponent. The log "undoes" the exponential.
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Exponential of a Logarithm

alog⁡a(x)=xa^{\log_a(x)} = x
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Applying an exponential to its own base's logarithm cancels both operations, returning the argument. The exponential "undoes" the log.
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Identities & Factoring

(11 formulas)

Difference of Squares

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)
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A difference of two perfect squares factors into the product of a sum and a difference. This is the most frequently used factoring identity in algebra.
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Square of a Sum

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
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Squaring a binomial sum produces a trinomial: the square of each term plus twice their product.
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Square of a Difference

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
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Squaring a binomial difference produces a trinomial: the square of each term minus twice their product. The result is always positive — a squared quantity.
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Cube of a Sum

(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
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Cubing a binomial sum. The coefficients follow the third row of Pascal's triangle: 1, 3, 3, 1.
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Cube of a Difference

(a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
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Cubing a binomial difference. Same coefficients as the cube of a sum, with alternating signs.
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Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
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A sum of two cubes factors into a binomial times a trinomial. The trinomial factor a2−ab+b2a^2 - ab + b^2 is irreducible over the reals.
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Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)
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A difference of two cubes factors into a binomial times a trinomial. Compare with the sum of cubes — the signs alternate in a predictable pattern: same, opposite, always positive.
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Trinomial Factoring Pattern

x2+(a+b)x+ab=(x+a)(x+b)x^2 + (a + b)x + ab = (x + a)(x + b)
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A monic quadratic trinomial factors into two binomials whose constants multiply to the constant term and add to the linear coefficient. This reverses the FOIL multiplication pattern.
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General Difference of Even Powers

x2n−a2n=(xn−an)(xn+an)x^{2n} - a^{2n} = (x^n - a^n)(x^n + a^n)
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Any difference of even powers splits as a difference of squares applied to the nn-th powers. Each factor may be factorable further depending on whether nn is even or odd.
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General Difference of Powers (odd n)

xn−an=(x−a)(xn−1+axn−2+⋯+an−1)x^n - a^n = (x - a)(x^{n-1} + ax^{n-2} + \cdots + a^{n-1})
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For any positive integer nn, the difference xn−anx^n - a^n has (x−a)(x - a) as a factor. The second factor is the sum of all terms xn−1−kakx^{n-1-k}a^k for k=0k = 0 to n−1n - 1.
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General Sum of Powers (odd n)

xn+an=(x+a)(xn−1−axn−2+⋯+an−1)x^n + a^n = (x + a)(x^{n-1} - ax^{n-2} + \cdots + a^{n-1})
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For odd nn, the sum xn+anx^n + a^n has (x+a)(x + a) as a factor. The second factor has alternating signs. This identity does not hold for even nn — a sum of even powers does not factor over the reals.
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Exponent Rules

(8 formulas)

Product Rule (Exponents)

am⋅an=am+na^m \cdot a^n = a^{m+n}
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Multiplying powers with the same base: keep the base, add the exponents.
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Quotient Rule (Exponents)

aman=am−n\frac{a^m}{a^n} = a^{m-n}
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Dividing powers with the same base: keep the base, subtract the exponents.
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Power of a Power

(am)n=amn(a^m)^n = a^{mn}
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Raising a power to another power: keep the base, multiply the exponents.
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Power of a Product

(ab)n=an⋅bn(ab)^n = a^n \cdot b^n
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A power applied to a product distributes to each factor.
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Power of a Quotient

(ab)n=anbn\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
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A power applied to a fraction distributes to numerator and denominator separately.
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Zero Exponent

a0=1(a≠0)a^0 = 1 \quad (a \neq 0)
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Any nonzero base raised to the zero power equals 1. This follows from the quotient rule: an/an=an−n=a0=1a^n / a^n = a^{n-n} = a^0 = 1.
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Negative Exponent

a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \quad (a \neq 0)
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A negative exponent means the reciprocal of the positive power. It does not make the result negative.
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Negative Exponent Flip

(ab)−n=(ba)n\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n
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A negative exponent on a fraction inverts the fraction and makes the exponent positive.
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Radical Rules

(7 formulas)

Radical to Exponent Conversion

an=a1/n\sqrt[n]{a} = a^{1/n}
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The nn-th root of aa equals aa raised to the power 1/n1/n. This bridges radical notation and exponential notation, allowing all radical operations to be performed using exponent rules.
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Product Rule (Radicals)

abn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}
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The nn-th root of a product equals the product of the nn-th roots. Used to simplify radicals by factoring the radicand.
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Quotient Rule (Radicals)

abn=anbn\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
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The nn-th root of a quotient equals the quotient of the nn-th roots.
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Power Rule (Radicals)

amn=am/n\sqrt[n]{a^m} = a^{m/n}
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Combines the radical to exponent conversion with the power of a power rule. The nn-th root of ama^m equals aa raised to the fraction m/nm/n.
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Nested Radicals

anm=amn\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}
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A radical inside a radical simplifies by multiplying the indices. In exponent form: (a1/n)1/m=a1/(mn)(a^{1/n})^{1/m} = a^{1/(mn)}.
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Even Root Identity

ann=∣a∣(n even)\sqrt[n]{a^n} = |a| \quad (n \text{ even})
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When the index is even, the nn-th root of ana^n returns the absolute value of aa, not aa itself. This is because even powers erase the sign: (−3)2=9(-3)^2 = 9, and 9=3\sqrt{9} = 3, not −3-3.
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Odd Root Identity

ann=a(n odd)\sqrt[n]{a^n} = a \quad (n \text{ odd})
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When the index is odd, the nn-th root of ana^n returns aa directly — no absolute value needed. Odd roots preserve sign.
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Polynomial Theorems

(8 formulas)

Remainder Theorem

P(x)=(x−c)⋅Q(x)+P(c)P(x) = (x - c) \cdot Q(x) + P(c)
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When a polynomial P(x)P(x) is divided by (x−c)(x - c), the remainder equals P(c)P(c) — the value of the polynomial evaluated at cc. No long division needed to find the remainder.
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Factor Theorem

(x−c) is a factor of P(x)  ⟺  P(c)=0(x - c) \text{ is a factor of } P(x) \iff P(c) = 0
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(x−c)(x - c) divides P(x)P(x) evenly if and only if cc is a root of PP. This is the remainder theorem with remainder equal to zero.
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Rational Root Theorem

If pq is a root: p∣a0 and q∣an\text{If } \frac{p}{q} \text{ is a root: } p \mid a_0 \text{ and } q \mid a_n
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For a polynomial with integer coefficients, any rational root p/qp/q (in lowest terms) must have pp dividing the constant term a0a_0 and qq dividing the leading coefficient ana_n. This limits the search for rational roots to a finite list of candidates.
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Vieta's Formulas (Quadratic)

x1+x2=−bax1⋅x2=cax_1 + x_2 = -\frac{b}{a} \qquad x_1 \cdot x_2 = \frac{c}{a}
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Relates the roots of a quadratic ax2+bx+c=0ax^2 + bx + c = 0 to its coefficients without solving the equation. The sum of the roots equals −b/a-b/a and the product equals c/ac/a.
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Binomial Theorem

(x+y)n=∑k=0n(nk)xn−kyk(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k
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Expands any positive integer power of a binomial as a sum of n+1n + 1 terms. Each term is weighted by a binomial coefficient (nk)\binom{n}{k}, with the powers of xx decreasing and the powers of yy increasing.
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Binomial Coefficient

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}
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Counts the number of ways to choose kk items from nn items, and gives the coefficient of the kk-th term in the binomial expansion. Read "nn choose kk."
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Pascal's Rule

(nk)+(nk+1)=(n+1k+1)\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1}
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Each entry in Pascal's triangle equals the sum of the two entries directly above it. This recurrence builds binomial coefficients row by row without computing factorials.
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Vieta's Formulas (General)

r1+r2+r3=−an−1r1r2r3=(−1)na0r_1 + r_2 + r_3 = -a_{n-1} \qquad r_1 r_2 r_3 = (-1)^n a_0
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Generalizes Vieta's formulas to polynomials of any degree (shown here for degree 3 with leading coefficient 1). The sum of the roots equals the negative of the second coefficient. The product of the roots equals the constant term times (−1)n(-1)^n.
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Arithmetic Sequences
Common DifferenceGeneral Term (Arithmetic Sequence)Recursive Form (Arithmetic Sequence)Arithmetic Series SumArithmetic MeanArithmetic Mean Property
Geometric Sequences
Common RatioGeneral Term (Geometric Sequence)Recursive Form (Geometric Sequence)Finite Geometric Series SumInfinite Geometric Series SumGeometric MeanGeometric Mean Property
Harmonic Sequences
General Term (Harmonic Sequence)Harmonic MeanAM-GM-HM Inequality
Recursive Sequences
Fibonacci RecurrenceGolden RatioBinet's FormulaCassini's IdentitySum of Fibonacci NumbersSum of Squared Fibonacci NumbersFibonacci GCD IdentityLucas RecurrenceLucas-Fibonacci RelationLucas Binet FormulaPrime Number Theorem
Figurate Numbers
Triangular Number FormulaRecursive Form (Triangular Numbers)Triangular Number as Binomial CoefficientSum of Consecutive Triangular NumbersSum of Triangular NumbersSquare Number FormulaRecursive Form (Square Numbers)Sum of First n Odd NumbersSum of SquaresPythagorean Triple Generator
Equations
Quadratic FormulaDiscriminantSquare Root PropertyCompleting the SquareAbsolute Value EquationAbsolute Value Inequalities
Logarithm Rules
Product Rule (Logarithms)Quotient Rule (Logarithms)Power Rule (Logarithms)Change of Base FormulaLogarithm of the BaseLogarithm of OneLogarithm of an ExponentialExponential of a Logarithm
Identities & Factoring
Difference of SquaresSquare of a SumSquare of a DifferenceCube of a SumCube of a DifferenceSum of CubesDifference of CubesTrinomial Factoring PatternGeneral Difference of Even PowersGeneral Difference of Powers (odd n)General Sum of Powers (odd n)
Exponent Rules
Product Rule (Exponents)Quotient Rule (Exponents)Power of a PowerPower of a ProductPower of a QuotientZero ExponentNegative ExponentNegative Exponent Flip
Radical Rules
Radical to Exponent ConversionProduct Rule (Radicals)Quotient Rule (Radicals)Power Rule (Radicals)Nested RadicalsEven Root IdentityOdd Root Identity
Polynomial Theorems
Remainder TheoremFactor TheoremRational Root TheoremVieta's Formulas (Quadratic)Binomial TheoremBinomial CoefficientPascal's RuleVieta's Formulas (General)