AlgebraAMC 8

Arithmetic Series

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

$n$ evenly spaced terms with first term $a_1$ and last term $a_n$ .

What it says

An arithmetic series adds numbers that go up by the same step each time. Its sum is the number of terms times the average term, and in an evenly spaced list the average is just the average of the first and last terms. The hardest part is usually counting the terms correctly.

Show proof

Write the sum forwards and backwards: $S = a_1 + \cdots + a_n$ and $S = a_n + \cdots + a_1$ . Each of the $n$ column pairs adds to $a_1 + a_n$ , so $2S = n(a_1 + a_n)$ .

When to use it

Summing evenly spaced numbers, like consecutive odd numbers or multiples of $7$ in a range.

Example

Find the sum of all odd integers from $11$ to $99$ , inclusive.

Show solution

There are $45$ terms, so the sum is $\frac{45(11+99)}{2} = 2475$ . Answer: $2475$