Arithmetic Series
In short: An arithmetic series is what you get by adding up an arithmetic sequence, whose terms climb or drop by a fixed step; the total is the number of terms times the mean of the two end terms.
Algebra 2AlgebraA-SSE.B.4
What is Arithmetic Series?
An arithmetic series is what you get when you add up a list of numbers that step by a fixed amount, like 14, 17, 20, 23. The list itself is a sequence; the sum is the series.
With first term \(a_1\), common difference \(d\) and \(n\) terms, the last term is \(a_n = a_1 + (n – 1)d\). Adding the first \(n\) terms gives \(S_n = \frac{n(a_1 + a_n)}{2}\). It works since every pair taken from opposite ends of the list, first with last, second with next-to-last, has an identical total.
How to spot it
- A story stacks things in rows or layers where each row has a fixed number more than the one before.
- The terms of a list increase by the same amount, and the problem asks for the total of that arithmetic series.
- Notation such as S_20 or a sigma sum of a linear expression in k appears.
- Adding by hand would take dozens of steps, so a shortcut is clearly needed.
The method
- Find the first term \(a_1\), the common difference \(d\) and the number of terms \(n\).
- Compute the last term with \(a_n = a_1 + (n – 1)d\) if it is not given.
- Substitute into \(S_n = \frac{n(a_1 + a_n)}{2}\).
- Finish the arithmetic and name the unit, such as musicians or dollars.
Arithmetic Series: solved example
Problem
A high school marching band lines up in 14 rows for a halftime shape. Row one holds 5 musicians and every later row holds 2 more than the row in front. Count the band.
Solution
Read off \(a_1 = 5\), \(d = 2\) and \(n = 14\). The back row therefore has \(a_{14} = 5 + 13(2) = 31\) players.
Total: \(S_{14} = \frac{14(5 + 31)}{2} = 7 \times 36 = 252\). The band has 252 musicians.
Try it
Question 1
Find the sum \(1 + 3 + 5 + \cdots + 79\), the first 40 odd numbers.
Show the answer
\(S_{40} = \frac{40(1 + 79)}{2} = 1{,}600\).
Question 2
Find \(S_{12}\) when \(a_1 = -5\) and \(d = 4\).
Show the answer
\(a_{12} = -5 + 11(4) = 39\), so \(S_{12} = \frac{12(-5 + 39)}{2} = 204\).
Arithmetic Series vs. Geometric Series
Here a fixed amount is added at every step. Over in the geometric kind, a fixed multiplier is applied instead. The list 5, 10, 15, 20 adds 5 each time, whereas 5, 10, 20, 40 doubles each time, so the two sums need different formulas.
Keep going
Lessons
Worksheets and quizzes
- Arithmetic and Geometric Sequences and Series – Worksheet
- Arithmetic and Geometric Sequences and Series – Quiz
- Sequences and Series – Worksheet
- Sequences and Series – Quiz
Related words
Whole dictionary: A–Z · Algebra 2 · Algebra