Algebra 2
Sequences and series
Arithmetic sequences and series
20 practice questions
0 video lessons
Theory + worked examples
Theory
An arithmetic sequence adds a constant common difference \(d\) each step:
\[a_n=a_1+(n-1)d,\qquad S_n=\dfrac{n}{2}(a_1+a_n).\]
A series is the sum of a sequence's terms.
The terms lie on a line — arithmetic growth is linear.
Terms rise by \(d=4\) each step.
The arithmetic formulas.
nth term and sum:
\[a_n=a_1+(n-1)d,\qquad S_n=\dfrac{n}{2}(a_1+a_n)\]
The sum formula is \(n\) times the average of the first and last terms.
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.
Example 1 — nth term
Find the \(10\)th term of \(3,7,11,15,\dots\).
Solution
Here \(a_1=3,\ d=4\).
| \(a_{10}\) | \(=\) | \(3+(10-1)(4)\) |
| \(=\) | \(3+36=39\) |
Example 2 — Sum of terms
Find the sum of the first \(10\) terms of \(3,7,11,\dots\).
Solution
Use \(S_n=\dfrac{n}{2}(a_1+a_n)\) with \(a_{10}=39\).
| \(S_{10}\) | \(=\) | \(\dfrac{10}{2}(3+39)\) |
| \(=\) | \(5(42)=210\) |
Example 3 — Find the difference
An arithmetic sequence has \(a_1=5\) and \(a_4=17\). Find \(d\).
Solution
Use \(a_4=a_1+3d\).
| \(17\) | \(=\) | \(5+3d\) |
| \(d\) | \(=\) | \(4\) |
Example 4 — Gauss sum
Find \(1+2+3+\dots+100\).
Solution
Arithmetic with \(a_1=1,\ a_{100}=100\).
| \(S_{100}\) | \(=\) | \(\dfrac{100}{2}(1+100)\) |
| \(=\) | \(50(101)=5050\) |
Common pitfalls
Use \((n-1)\), not \(n\), in the term formula.
The sum needs the last term \(a_n\) — find it first.
A common difference is added, not multiplied.
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.
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