# Variation problem

• Dec 2nd 2010, 12:34 AM
jgv115
Variation problem
The sum of the first $n$ natural numbers is equal to the sum of two quantities, the first of which is proportional to $n$ and the second to $n^2$. Work out the sums of the first three and four natural numbers and hence find the formula for the sum of the first $n$ natural numbers.

How do I do this?
• Dec 2nd 2010, 01:33 AM
Wilmer
Quote:

Originally Posted by jgv115
The sum of the first $n$ natural numbers is equal to the sum of two quantities, the first of which is proportional to $n$ and the second to $n^2$. Work out the sums of the first three and four natural numbers and hence find the formula for the sum of the first $n$ natural numbers.

Can you follow this:

n = 4; sum 1 to n = 1+2+3+4 = 10
kn^2 + kn = 10
16k + 4k = 10
k = 1/2

(1/2)n^2 + (1/2)n = 10
(n^2 + n) / 2 = 10
n(n+1) / 2 = 10 = sum