The sum of the first n odd numbers is always n squared

Add the first few odd numbers and watch the totals:

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

The totals are 1, 4, 9, 16 and 25, which are the perfect squares 1², 2², 3², 4² and 5². The sum of the first n odd numbers is n². This is true for every n, and you can see why with a picture.

The picture: squares built from L-shapes

Start with one dot. To make a 2 by 2 square of dots, add an L-shaped border of 3 dots. To grow it to 3 by 3, add an L of 5 dots. Then an L of 7 dots makes 4 by 4, and an L of 9 dots makes 5 by 5. Every new L has two more dots than the one before it, so the L-shapes contain exactly the odd numbers 1, 3, 5, 7, 9. Stacking them up builds a square, so adding the odd numbers gives a square number.

The proof with algebra

Suppose the first n odd numbers add up to n². The next odd number is 2n + 1 (for n = 4, the next odd number after 1, 3, 5, 7 is 9, and 2 × 4 + 1 = 9). Adding it:

n² + (2n + 1) = (n + 1)²
The right side is the identity (a + b)² = a² + 2ab + b² from Class 8, with a = n and b = 1.

So if the rule holds for n odd numbers, it holds for n + 1. It holds for n = 1 (the sum is 1 = 1²), so it holds for 2, then 3, then every number after that.

The Class 10 proof

The odd numbers 1, 3, 5, ... form an arithmetic progression with first term a = 1 and common difference d = 2. The sum formula from the AP chapter is S = n/2 × [2a + (n - 1)d]. Substitute:

S = n/2 × [2 + 2(n - 1)]
= n/2 × 2n
= n²

The same answer from a completely different route, which is always a good sign.

Using it

1 + 3 + 5 + ... + 19 has 10 terms, so the sum is 10² = 100.
1 + 3 + 5 + ... + 99 has 50 terms, so the sum is 50² = 2500.
To find how many terms there are, use the last odd number: the count is (last + 1) / 2. For 99 that is 100 / 2 = 50.
You can also go backwards: 169 is 13², so it is the sum of the first 13 odd numbers.

Try it yourself

Find 1 + 3 + 5 + ... + 29.
Find 1 + 3 + 5 + ... + 49, and then 51 + 53 + ... + 99. (Hint: subtract.)
Which odd numbers add up to 144, starting from 1?

Answers: the first sum has 15 terms, so it is 225. The second pair gives 625 and then 2500 - 625 = 1875. Since 144 = 12², the sum of the first 12 odd numbers is 144. Arithmetic progressions are covered in Class 10, and you can start with the Class 10 chapters. When you want practice, build a fresh test with the Practice Generator.