Question

The number 2016 can be written as the sum of 21 consecutive whole numbers. What is the sum of the 16th number and the 20th number?

Answer 1

If x is the lowest whole number then the others are x+1, x+2, ..., x+20.

Their sum can be written:

(x+x+20)+(x+1+x+19)+(x+2+x+18)+...+(x+9+x+11)+(x+10)=2016.

That is, 10(2x+20)+x+10=20x+200+x+10=2016, or 21x+210=2016.

Therefore 21x=2016-210=1806, x=1806/21=86.

So the numbers are 86, 87, 88, ..., 106.

The 16th number is x+15=101 and the 20th is x+19=105, so their sum is 206.