# Shortcuts for Adding a Series

> Pair the first and last numbers to add long lists fast, and count how many numbers are in a list.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/adding-a-series/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The boy who would not do it the slow way
More than 200 years ago in Germany, a teacher wanted some peace and quiet. He told his class, "Add up every number from 1 to 100." He expected it to take an hour.

A young boy named Carl Gauss put his hand up after a minute with the answer: 5050.

Carl had noticed that $1 + 100 = 101$, $2 + 99 = 101$, $3 + 98 = 101$, and so on. There are 50 such pairs, so the total is $50 \times 101 = 5050$. You are about to learn his trick.

**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.

*Diagram: The even numbers 2 to 20 written in a row, with arcs joining 2 and 20, 4 and 18, 6 and 16, 8 and 14, 10 and 12; each pair adds to 22*
*Five pairs, each adding to 22, so the total is 5 times 22, which is 110.*

### Worked example: Even numbers from 2 to 20
$(2 + 20) + (4 + 18) + (6 + 16) + (8 + 14) + (10 + 12)$

There are 5 pairs and each makes 22. Total $= 5 \times 22 = 110$.

## Step 1: count the numbers

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

### Key idea: Counting a list
$$\text{how many} = \frac{\text{last} - \text{first}}{\text{gap}} + 1$$

The **gap** is the step between numbers: 1 for all numbers, 2 for odd or even numbers.

### Worked example: How many?
Even numbers from 14 to 40: $\frac{40 - 14}{2} + 1 = 13 + 1 = 14$ numbers.

All numbers from 50 to 80: $\frac{80 - 50}{1} + 1 = 31$ numbers.

### Watch out: Do not forget the +1
From 1 to 10 there are 10 numbers, but $10 - 1 = 9$. That is why we add 1. It is the fence-post problem: 10 posts have only 9 gaps between them.

## Step 2: pair and multiply

### Key idea: The shortcut
$$\text{total} = \frac{(\text{first} + \text{last}) \times \text{how many}}{2}$$

This works whether there is an even or an odd number of terms.

### Worked example: An even number of terms
Even numbers from 14 to 40. There are 14 numbers, so 7 pairs. Each pair: $14 + 40 = 54$.

Total $= 7 \times 54 = 378$.

### Worked example: An odd number of terms
All numbers from 1 to 15. There are 15 numbers: 7 pairs that each make 16, plus the middle number 8 left alone.

Total $= 7 \times 16 + 8 = 120$.

The formula agrees: $\frac{16 \times 15}{2} = 120$.

### Exam tip: The middle number trick
When the count is odd, the middle number is the **average**. Total $=$ middle $\times$ how many. Numbers 20 to 30: middle is 25, count is 11, total $= 25 \times 11 = 275$.

### Challenge: Selective level: odd numbers make squares
$1 = 1$, $1 + 3 = 4$, $1 + 3 + 5 = 9$, $1 + 3 + 5 + 7 = 16$.

Adding the first $n$ odd numbers gives $n \times n$.

A bus picks up 1 passenger at stop 1, 3 at stop 2, 5 at stop 3, and so on (nobody gets off). At stop 12 it picks up $2 \times 12 - 1 = 23$ people. After 10 stops there are $10 \times 10 = 100$ people on board, and after 15 stops there are 225.

**Mind map**

```text
Adding a series
  Count first
    (last - first) ÷ gap + 1
    Remember the +1
  Pair up
    First + last
    Every pair is the same
  Formula
    (first + last) x count ÷ 2
  Odd count
    Middle number is left over
    Middle x count
  Odd numbers
    $1 + 3 + 5 = 3^2$
```

## 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 $= (\text{first} + \text{last}) \times \text{how many} \div 2$.
- 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

Try these questions from the chapter quiz.

1. What is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10$?
   - A. 50
   - B. 55
   - C. 100
   - D. 110
   - Answer: B. 55. 5 pairs of 11: $5 \times 11 = 55$.

2. How many numbers are there from 1 to 20 inclusive?
   - A. 19
   - B. 20
   - C. 21
   - D. 10
   - Answer: B. 20. $\frac{20 - 1}{1} + 1 = 20$.

3. Add the even numbers from 2 to 20 inclusive.
   - A. 110
   - B. 100
   - C. 220
   - D. 90
   - Answer: A. 110. 5 pairs, each making 22: $5 \times 22 = 110$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/adding-a-series/#quiz