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Geometric sequence and series

In a geometric sequence you multiply by the same ratio for each new term. The $n$ th term is the first term times the ratio raised to $n - 1$ . The sum of the first $n$ terms has its own formula, which applies when the ratio is not 1. You can check by writing out the terms and adding them. A constant factor, not a constant difference, 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

With $a_1 = 2$ and $k = 3$ , $a_4 = 2\cdot 3^3 = 54$ . The sum of the first four is $2\cdot\dfrac{3^4 - 1}{3 - 1} = 80$ .

The exponent in term number

n

n - 1

. In the sum formula the exponent is

n

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Part of Mathematics R2: Sequences and series.