Shortcuts for Adding a Series

In short: To add a long list of evenly spaced numbers quickly, pair the first with the last, the second with the second last, and so on: every pair has the same total. So the total is (first + last) × how many ÷ 2.

Before you start

• Even numbers end in 0, 2, 4, 6 or 8. Odd numbers end in 1, 3, 5, 7 or 9.
• Inclusive means the first and last numbers are included. "From 2 to 20 inclusive" includes 2 and 20.
• Multiplying 2-digit numbers, like $7\times 54$.

The pairing trick

Write the list, then join the first and last, the second and second last, and so on. Every pair adds to the same total.

Step 1: count the numbers

Before you pair, you must know how many numbers are in the list.

Step 2: pair and multiply

Quick summary

• Count the numbers first: $\frac{\text{last}-\text{first}}{\text{gap}}+1$.
• Pair first with last, second with second last. Every pair has the same total.
• Total $=(\text{first}+\text{last})\times \text{how many}÷2$.
• With an odd count, the middle number is left over. It is also the average.
• The first $n$ odd numbers add up to $n^2$.

Quiz