Arithmetic series: nth term and partial sums
Arithmetic Series
Arithmetic Series is a topic in Sequences & Series in the California Common Core State Standards. It is aligned to Standard A-SSE.4, which requires students to derive and use the formula for the sum of an arithmetic series.
An arithmetic series sums a sequence with a constant difference, using \(a_n=a_1+(n-1)d\) and \(S_n=\dfrac{n}{2}(a_1+a_n)\).
Theory
An arithmetic sequence has a constant common difference \(d\) between consecutive terms. Its \(n\)th term is
The sum of the first \(n\) terms, an arithmetic series, is
The nth term and the partial sum:
How to work with arithmetic series
- Find \(d\) as the difference of consecutive terms.
- nth term: \(a_n=a_1+(n-1)d\).
- Sum: \(\dfrac{n}{2}(a_1+a_n)\), or the expanded form if \(a_n\) is unknown.
- Solve for \(n\) from a term or a sum when required.
\(a_1=3,\ d=4\); use \(a_n=a_1+(n-1)d\).
| a_{20} | \(=\) | 3+(20-1)4 |
| \(=\) | 3+76=79 |
Use \(S_n=\dfrac{n}{2}(a_1+a_n)\) with \(a_{20}=79\).
| S_{20} | \(=\) | \dfrac{20}{2}(3+79) |
| \(=\) | 10\cdot 82=820 |
Use \(S_n=\dfrac{n}{2}(2a_1+(n-1)d)\).
| S_{15} | \(=\) | \dfrac{15}{2}(2\cdot 5+14\cdot 2) |
| \(=\) | \dfrac{15}{2}(38)=285 |
Solve \(a_n=35\) with \(a_1=2,\ d=3\).
| 2+(n-1)3 | \(=\) | 35 |
| (n-1)3 | \(=\) | 33 |
| n | \(=\) | 12 |
Common pitfalls
Frequently asked questions
What is an arithmetic sequence?
A sequence with a constant difference \(d\) between consecutive terms; the \(n\)th term is \(a_1+(n-1)d\).
How do you find the sum of an arithmetic series?
\(S_n=\dfrac{n}{2}(a_1+a_n)\) — the average of the first and last term times the number of terms.
What is the common difference?
The fixed amount \(d\) added to each term to get the next.
Why is it (n - 1)d and not nd?
Because the first term needs no addition; only \(n-1\) steps of \(d\) reach the \(n\)th term.