Consecutive numbers

In your book, Maths Mela (Chapter 4, We the Travellers — II), numbers that follow one another in order without skipping any number are called consecutive numbers. 1, 2, 3, 4, 5 are consecutive numbers. So are 29, 30, 31, 32, and 512, 513.

The book adds 2 consecutive numbers: 1 + 2 = 3, 2 + 3 = 5, 3 + 4 = 7, 4 + 5 = 9. Every sum is odd! That is because the second number is always 1 more than the first, so the sum is a double plus 1, and a double plus 1 is always odd. Each sum is 2 more than the one before.

Now 3 consecutive numbers: 1 + 2 + 3 = 6, 2 + 3 + 4 = 9, 3 + 4 + 5 = 12, 4 + 5 + 6 = 15. Each sum is 3 more than the one before, because each of the 3 numbers has gone up by 1. And 4 consecutive numbers: 1 + 2 + 3 + 4 = 10, 2 + 3 + 4 + 5 = 14, 3 + 4 + 5 + 6 = 18, 4 + 5 + 6 + 7 = 22. These sums are always even, and each is 4 more than the one before. The sums go up by the number of numbers being added.

Look at 3 + 4 + 5 = 12. The middle number is 4, and 3 × 4 = 12. The 3 is 1 less than the middle and the 5 is 1 more, so together they are like two middle numbers. So the sum of 3 consecutive numbers is 3 times the middle number: 67 + 68 + 69 = 3 × 68 = 204.

The same works for 5 consecutive numbers: the sum is 5 times the middle number. 2 + 3 + 4 + 5 + 6 = 5 × 4 = 20, and 48 + 49 + 50 + 51 + 52 = 5 × 50 = 250.