Turn it upside down

The idea

Writing 1 + 2 + … + n forwards and backwards and adding gives 2S = n(n + 1), so Sₙ = n(n + 1)/2, which is also the nth triangular number.

Add 1 to 10

Can you find 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 without adding all ten numbers one by one?

Roughly, what do you think the total is?

• About 45
• About 55
• About 100

Make a rectangle

The orange dots show 1 + 2 + … + n. Turn a copy upside down and fit it in. What shape do you get?

Forwards and backwards

Writing 1 + 2 + … + n forwards and backwards and adding gives 2S = n(n + 1), so Sₙ = n(n + 1)/2. This is also the nth triangular number.

Write S = 1 + 2 + … + 10 and S = 10 + 9 + … + 1. Each pair adds up to 11, and there are 10 pairs, so 2S = 110 and S = 55.

Notes

1 + 2 + … + n written forwards and backwards and added gives 2S = n(n + 1), so Sₙ = n(n + 1)/2, also the nth triangular number.

Check yourself

What is the sum of the first 20 natural numbers? (S₂₀)

Answer: 210

S₂₀ = 20 × 21 ÷ 2 = 420 ÷ 2 = 210.

Find 1 + 2 + 3 + … + 100.

Answer: 5050

S₁₀₀ = 100 × 101 ÷ 2 = 10100 ÷ 2 = 5050.

Why do we divide n(n + 1) by 2 at the end?

What is the 12th triangular number?

Answer: 78

12 × 13 ÷ 2 = 156 ÷ 2 = 78.

• Because the rectangle holds the sum twice: once forwards and once backwards — correct. Yes. 2S = n(n + 1), so S is half of it.
• Because there are 2 numbers in the sum. The sum has n numbers. We divide by 2 because we added the sum to a reversed copy of itself.
• Because n is always even. n can be odd or even. n(n + 1) is always even, and half of it is the sum.