Algebra 2 / Unit 7: Sequences and Series
7.4 Arithmetic Series
Notes
A series is the sum of the terms of a sequence. For an arithmetic series:
$$S_n = \frac{n(a_1 + a_n)}{2}$$
- In words: number of terms times the average of the first and last.
- If you don't know $a_n$, find it with $a_n = a_1 + (n - 1)d$ first.
- Count terms with $\displaystyle n = \frac{a_n - a_1}{d} + 1$.
- Example: $\displaystyle 1 + 2 + \dots + 100 = \frac{100(101)}{2} = 5050$.
Practice
Find the sum of the first 50 positive integers.
$\displaystyle \frac{50(1 + 50)}{2} = 1275$.
Find $5 + 9 + 13 + \dots + 81$.
$\displaystyle n = \frac{81 - 5}{4} + 1 = 20$ terms, so $\displaystyle S = \frac{20(86)}{2} = 860$.
A theater has 15 rows. The first row has 20 seats and each row has 2 more than the row before. How many seats in all?
Last row: $20 + 14(2) = 48$. $\displaystyle S = \frac{15(20 + 48)}{2} = 510$.
Ready for more? Take the Unit 7 test.