Geometric Sequences and Series

Starting point: Years 9 and 10

Multiplying patterns

In 3, 6, 12, 24, ..., each term is double the one before. To find the multiplier, divide a term by the one before it: $\frac{6}{3}=2$.

In 80, 40, 20, 10, ..., the multiplier is $\frac{40}{80}=\frac{1}{2}$, so the terms halve.

Adding or multiplying?

Repeated percentage change

Growing by 10% means multiplying by 1.1 each step. A 1000 dollar investment growing 10% a year gives 1000, 1100, 1210, 1331, ..., a geometric sequence with multiplier 1.1.

Quick summary

• A geometric sequence multiplies by the same ratio $r$ each step: $r=\frac{T_2}{T_1}$.
• $T_n=a{r}^{n-1}$.
• $S_n=\frac{a(r^n-1)}{r-1}$.
• If $-1<r<1$, the infinite sum is $S_{\infty }=\frac{a}{1-r}$.
• Repeated percentage change gives a geometric sequence.

Quiz