Add every third integer from 7 to 148 inclusive. State the count of terms as well as the sum.

Add every third integer from 7 to 148 inclusive. State the count of terms as well as the sum.

Approach: Count the terms with the fencepost formula on the common difference, then multiply the count by the mean of the two endpoints.

3720. The run is an arithmetic series with first term 7, common difference 3 and last term 148, so the count is (148 - 7)/3 + 1 = 47 + 1 = 48 terms. The sum is the count times the mean of the endpoints: 48 * (7 + 148)/2 = 48 * 77.5 = 24 * 155 = 3720. Pairing the ends gives the same result, since each of the 24 pairs sums to 155. The fencepost count is where errors happen under a clock, so always divide the range by the step and add 1, then sanity check that 148 - 7 is 141, which is divisible by 3, so 148 really is a member of the run.

Follow-up: What is the sum of every third integer from 7 to 148 that is also even?

Key concepts: arithmetic series, common difference, fencepost count.