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Algebra Formulas
Browse 6 reviewed algebra formulas with concrete inputs, outputs, calculations, applicability checks, and LaTeX.
Algebra
Arithmetic Sequence Formula
a_n=a_1+(n-1)dFinds the nth term of a sequence with a constant difference.
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Arithmetic Series Sum
S_n=\frac n2(a_1+a_n)Computes the sum of the first n terms of an arithmetic sequence.
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Distance Formula
d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}Calculates the straight-line distance between two points in a coordinate plane.
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Geometric Sequence Formula
a_n=a_1r^{n-1}Finds the nth term of a sequence with a constant ratio.
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Quadratic Formula
x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}Finds the roots of a quadratic equation ax² + bx + c = 0 when a is nonzero.
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Slope-Intercept Form
y=mx+bExpresses a straight line using its slope and vertical intercept.
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Algebra
Arithmetic Sequence Term
a_n=a_1+(n-1)dComputes the nth term of an arithmetic sequence.
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Binomial Theorem
(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^kExpands a nonnegative integer power of a binomial.
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Complex Number Modulus
|z|=\sqrt{a^2+b^2},\quad z=a+biCalculates the distance of a complex number from the origin in the complex plane.
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Complex Number Polar Form
z=r(\cos\theta+i\sin\theta)=re^{i\theta}Expresses a complex number by magnitude and argument.
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Euler Identity
e^{i\pi}+1=0Connects the constants e, i, π, one, and zero in a single complex exponential identity.
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Exponent Product Rule
a^m a^n=a^{m+n}Combines powers with the same base by adding exponents.
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Exponent Quotient Rule
\frac{a^m}{a^n}=a^{m-n}Divides powers with the same nonzero base by subtracting exponents.
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Finite Geometric Series Sum
S_n=a_1\frac{1-r^n}{1-r}Computes a finite geometric series when the ratio is not one.
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Geometric Sequence Term
a_n=a_1r^{n-1}Computes the nth term of a geometric sequence.
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Harmonic Number
H_n=\sum_{k=1}^{n}\frac1kDefines the nth harmonic number.
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Infinite Geometric Series Sum
S=\frac{a_1}{1-r},\quad |r|<1Computes the sum of a convergent infinite geometric series.
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Logarithm Change of Base
\log_b x=\frac{\log_k x}{\log_k b}Rewrites a logarithm using another convenient base.
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Logarithm Product Rule
\log_b(xy)=\log_b x+\log_b yConverts a logarithm of a product into a sum of logarithms.
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Point-Slope Form
y-y_1=m(x-x_1)Defines a line from one point and its slope.
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Quadratic Discriminant
\Delta=b^2-4acDetermines the number and type of roots of a quadratic equation.
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Quadratic Vertex Form
y=a(x-h)^2+kExpresses a parabola using its vertex and vertical scale.
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Quadratic Vertex Formula
x_v=-\frac{b}{2a},\quad y_v=f(x_v)Finds the vertex coordinates of a quadratic function.
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Sum of the First Cubes
\sum_{k=1}^{n}k^3=\left[\frac{n(n+1)}2\right]^2Computes the sum of the first n cube numbers.
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Sum of the First Integers
\sum_{k=1}^{n}k=\frac{n(n+1)}2Computes the sum of the first n positive integers.
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Sum of the First Squares
\sum_{k=1}^{n}k^2=\frac{n(n+1)(2n+1)}6Computes the sum of the first n square numbers.
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