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Geometric sequence and series

In a geometric sequence you multiply by the same ratio for each new term. Term number n is the first term times the ratio raised to n minus 1. The sum of the first n terms has a formula that applies when the ratio is not 1. You can check by writing out the first terms and adding them. A constant factor is the sign to look for.

Try it yourself

Geometric series

a_n = a_1 k^{n-1}

term number n

S_n = a_1\frac{k^n - 1}{k - 1}

the sum

Symbols

$a_1$	the first term
$k$	the ratio

Example

$a_1 = 2$ and $k = 3$ give $a_5 = 2 \cdot 3^4 = 162$ .

The sum of the first five is $2 \cdot \dfrac{3^5 - 1}{3 - 1} = 242$ .

The exponent in term number n is n minus 1. In the sum formula it is n.

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Part of Mathematics S2: Sequences and series.