Two staircases make a rectangle

The idea

The sum of the first n positive integers is Sₙ = n(n + 1)/2.

Count the squares

A staircase has 1 square in the first row, 2 in the second, 3 in the third, and so on, up to n rows.

How many squares are there in all? That is 1 + 2 + 3 + … + n.

Make a second copy of the staircase, turn it upside down and fit it against the first. What shape do you get?

• A rectangle
• A bigger staircase
• A triangle

Flip a copy and fit it in

Choose the number of rows n. Press Flip a copy. Count the rectangle, then take half. Make a staircase of 15 squares, then one of 55.

The sum of the first n integers

Sₙ = n(n + 1)/2 is the sum of the first n positive integers.

Two staircases make a rectangle with n rows and n + 1 columns. So 2 × Sₙ = n(n + 1).

This is the sum formula S = (n/2)(a + *l*) with first term 1 and last term n. For Gauss's 100 numbers: S = 100 × 101 ÷ 2 = 5050.

Notes

Sₙ = n(n + 1)/2 is the sum 1 + 2 + 3 + … + n. Two staircases of n rows fit together into a rectangle n by (n + 1).

Check yourself

Find 1 + 2 + 3 + … + 20.

Answer: 210

S = 20 × 21 ÷ 2 = 420 ÷ 2 = 210.

Find the sum of the first 1000 positive integers.

Answer: 500500

S = 1000 × 1001 ÷ 2 = 500 × 1001 = 500500.

Two staircases of n rows make a rectangle. How many columns does it have?

A staircase has 66 squares in all. How many rows does it have?

Answer: 11

2 × 66 = 132 = 11 × 12, so n = 11. Check: 11 × 12 ÷ 2 = 66.

• n + 1 — correct. Yes! The first row has 1 square from the staircase and n from the flipped copy: 1 + n.
• n. The rectangle has n rows, but each row is one longer: 1 + n squares.
• 2n. Each row has 1 + n squares, not 2n.