Adding the first n

A hall has 20 rows. Row 1 has 1 chair, row 2 has 2, and so on. How many chairs?

The rule

The hall has 210 chairs. The sum of a list is its total.

Write the list twice, the second time backwards. Add each number to the one below it.

1 + 2 + 3 + 4 and 4 + 3 + 2 + 1. The pairs are 1 + 4, 2 + 3, 3 + 2 and 4 + 1. Each pair makes 5. There are 4 pairs.

Two copies make 4 × 5 = 20. One copy is half of that: 10.

Now take the first n numbers, 1 to n. Each pair makes n + 1, and there are n pairs. So one copy is half of n × (n + 1).

1 + 2 + … + n = n(n + 1) ÷ 2

For the hall, n = 20. 20 × 21 = 420, and 420 ÷ 2 = 210 chairs.

The sums 1, 3, 6, 10, 15, … are the triangular numbers. The nth one is n(n + 1) ÷ 2. Rows of 1, 2, 3, … dots make a triangle.

Try the lab to finish this lesson.