Skill: 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,

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