Calculate sums of arithmetic series

A finite arithmetic series is the sum of terms in an arithmetic sequence, whose consecutive terms differ by a constant common difference. The sum can be determined by pairing first and last terms, using $S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}[2a_1+(n-1)d]$ , while distinguishing the number of terms from the value of a term; infinite-series convergence and more advanced generalizations are outside this scope.

Detailed Explanation: Calculate sums of arithmetic series

An arithmetic series is the sum of the terms of an arithmetic sequence. The difference between consecutive terms is constant.

To calculate its sum, use

S_n=\frac{n}{2}(a_1+a_n),

where:

$S_n$ is the sum,
$n$ is the number of terms,
$a_1$ is the first term,
$a_n$ is the last term.

Example

Find the sum

7+11+15+\cdots+63.

Step 1: Identify the values.

The first term is

a_1=7,

and the common difference is

d=11-7=4.

The last term is

a_n=63.

We still need to find $n$ , the number of terms.

Step 2: Find the number of terms.

Use the formula for the $n$ th term:

a_n=a_1+(n-1)d.

Substitute the known values:

63=7+(n-1)(4).

Solve:

56=4(n-1)

14=n-1

n=15.

There are $15$ terms. Notice that $n$ is a count of terms, not the value of the last term.

Step 3: Calculate the sum.

Use

S_n=\frac{n}{2}(a_1+a_n).

Thus,

S_{15}=\frac{15}{2}(7+63)

S_{15}=\frac{15}{2}(70)

\boxed{S_{15}=525}.

So,

7+11+15+\cdots+63=525.

Learn by doing: Calculate sums of arithmetic series

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Sums - Series of Integers 1 to N - Addition to Sum

Earned ?