Arithmetic sequences and series
Arithmetic Sequences and Series
Arithmetic Sequences and Series is the opening topic of Sequences & Series in the Common Core State Standards. It is aligned to Standard F-BF.2, which requires students to write arithmetic sequences recursively and explicitly and find the sum of a finite series.
An arithmetic sequence adds a constant difference; its nth term is \(a_1+(n-1)d\) and the sum of \(n\) terms is \(\dfrac{n}{2}(a_1+a_n)\).
Theory
An arithmetic sequence adds a constant common difference \(d\) each step:
A series is the sum of a sequence's terms.
nth term and sum:
How to work with arithmetic sequences
- Find \(a_1\) and the common difference \(d\).
- Use \(a_n=a_1+(n-1)d\) for a term.
- Find \(a_n\) first, then \(S_n=\dfrac{n}{2}(a_1+a_n)\).
- Solve for an unknown using the term formula.
Here \(a_1=3,\ d=4\).
| \(a_{10}\) | \(=\) | \(3+(10-1)(4)\) |
| \(=\) | \(3+36=39\) |
Use \(S_n=\dfrac{n}{2}(a_1+a_n)\) with \(a_{10}=39\).
| \(S_{10}\) | \(=\) | \(\dfrac{10}{2}(3+39)\) |
| \(=\) | \(5(42)=210\) |
Use \(a_4=a_1+3d\).
| \(17\) | \(=\) | \(5+3d\) |
| \(d\) | \(=\) | \(4\) |
Arithmetic with \(a_1=1,\ a_{100}=100\).
| \(S_{100}\) | \(=\) | \(\dfrac{100}{2}(1+100)\) |
| \(=\) | \(50(101)=5050\) |
Common pitfalls
Frequently asked questions
What is an arithmetic sequence?
A sequence with a constant difference between terms.
What is the formula for the nth term?
\(a_n=a_1+(n-1)d\).
How do you sum an arithmetic series?
\(S_n=\dfrac{n}{2}(a_1+a_n)\).
What is the common difference?
The constant amount added to get the next term.