# Sum of k

How to solve the given summation by using the sum of k formula: formula, 1 example, and its solution.

## Formula∑_{k = 1}^{n} k

The sum of k, as k goes from 1 to n, is

n(n + 1)/2.

This means

1 + 2 + 3 + ... + n = n(n + 1)/2.

You can prove this formula

by using mathematical induction.

## Example∑_{k = 1}^{n} (4k - 3)

Solution

Solution (Detail)

The coefficient 4 gets out from the sigma.

k is in the sigma.

-3 is a constant.

So write -3n.

Summation: How to Solve

Write the coefficient 4.

The sum of k is

n(n + 1)/2.

And write -3n.

So 4⋅n(n + 1)/2 - 3n.

Cancel the denominator 2

and reduce the numerator 4 to, 4/2, 2.

2⋅n(n + 1) = 2n^{2} + 2n

+2n - 3n = -n

So

2n^{2} - n

is the answer.