What is the method for finding the sum of 10 consecutive numbers?

Answer 1

The question, as asked, has no answer since there is no such thing as a consecutive number. Given any two numbers there is an infinity of numbers between them so non can be consecutive. However, I assume you meant consecutive INTEGERS, though there is a huge difference between integers and numbers.

There are several methods. One of the simplest is as follows:

Suppose the numbers are x, x+1, x+2, ... The last of the 10 will be x+9

Let their sum be S

So S = x + (x+1) + (x+2) + ... + (x+9)

Rewriting the 10 numbers in reverse order,

S = (x+9) + (x+8) + (x+7) + ... + x

Take the first number on the right hand side of each equation. Their sum is x + (x+9) = x+10.

Take the second number on the right hand side of each equation. Their sum is (x+1) + (x+9) = x+10.

Take the third number on the right hand side of each equation. Their sum is (x+2) + (x+8) = x+10.

etc.

Thus you have 10 pairs of numbers and each pair sums to x+10.

So if you add the two equations, you get

2*S = 10*(x+10)

or S = 5*(x+10).

The result can be generalised as follows:

The sum of k consecutive integers, from x to x+k-1, is k*(x+k)/2