Geometric Sequences and Series

Sequences and seriesArithmetic Sequences and SeriesGeometric Sequences and SeriesSpecial Sums and Means

Terms

Sequence

a, a r, a r^{2}, a r^{3}, \dots

a ≠ 0, r ≠ 0

Common ratio

r = \frac{a_{2}}{a_{1}} = \frac{a_{n}}{a_{n - 1}}

nth term

a_{n} = a_{1} r^{n - 1} = a_{m} r^{n - m}

kth term from the end

\frac{l}{r^{k - 1}}

l the last term

Recursive rule

a_{n} = r \cdot a_{n - 1}

n ≥ 2

Sum of n terms

S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}

r ≠ 1

Sum from the last term

S_{n} = \frac{a_{1} - r l}{1 - r}

r ≠ 1

Sum when r = 1

S_{n} = n a_{1}

Infinite sum

S_{\infty} = \frac{a_{1}}{1 - r}

|r| < 1

Product of n terms

P_{n} = a_{1}^{n} r^{\frac{n (n - 1)}{2}} = (a_{1} a_{n})^{\frac{n}{2}}

Mean of two numbers

\sqrt{a b}

a, b > 0

Mean of n numbers

\sqrt[n]{x_{1} x_{2} \dots x_{n}}

Three terms in a row

a, b, c geometric \Leftrightarrow b^{2} = a c

k means between a and b

r = {(\frac{b}{a})}^{\frac{1}{k + 1}}

Product of the k means

{(\sqrt{a b})}^{k}

Three terms, centered

\frac{a}{r}, a, a r

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