Algebra 2 / Unit 7: Sequences and Series
7.5 Geometric Series
Notes
For a finite geometric series with $r \ne 1$:
$$S_n = \frac{a_1(1 - r^n)}{1 - r}$$
- $n$ is the number of terms, not the last term.
- Example: $\displaystyle 3 + 6 + 12 + 24 + 48 = \frac{3(1 - 2^5)}{1 - 2} = \frac{3(-31)}{-1} = 93$.
- This formula shows up in savings plans, loans and growth problems.
Practice
Find the sum of the first 6 terms of $2 + 6 + 18 + \dots$
$\displaystyle \frac{2(1 - 3^6)}{1 - 3} = \frac{2(-728)}{-2} = 728$.
Evaluate $\displaystyle \sum_{k=1}^{5} 4\left(\frac{1}{2}\right)^{k - 1}$.
$4 + 2 + 1 + 0.5 + 0.25 = 7.75$.
You save 1 cent on day 1 and double the amount each day. How much have you saved in total after 10 days?
$\displaystyle \frac{1(1 - 2^{10})}{1 - 2} = 1023$ cents. $5.12 is only the day-10 amount.
Ready for more? Take the Unit 7 test.