SL 1.3—Geometric sequences and series

Use the common ratio to model geometric growth.

A geometric sequence multiplies by the same ratio: aₙ=a₁rⁿ⁻¹; finite sums use repeated multiplication.

Worked example
For 5, 10, 20, r=2, so a₅=5·2⁴=80.

Worked example
Is 3, 6, 12 arithmetic or geometric? geometric, because each term is multiplied by 2.

Common boundary
Equal increases do not make a sequence geometric; check multiplication.

Complete finite-series example: for $3,6,12,24,48$, $a_1=3$, $r=2$ and $n=5$. $S_5=\frac{a_1(1-r^n)}{1-r}=\frac{3(1-2^5)}{1-2}=93$, equivalently $\sum_{k=1}^{5}3\cdot2^{k-1}=93$. This finite formula is valid for $r\ne1$; if $r=1$, $S_n=na_1$.