Question

Given that k is a positive integer. Find, in terms of k, an expression for Sn, which is the sum of the integers from 2k to 4k inclusive​

Answer 1

`S_n=3k\cdot (2k+1)=6k^2 +3k`

Explanation:

k is a positive integer.

Consider an arithmetic sequence:

`a_1=2k\\ \\a_n=4k\\ \\d=1`

First, find n:

`a_n=a_1+(n-1)\cdot d\\ \\4k=2k+(n-1)\cdot 1\\ \\2k=n-1\\ \\n=2k+1`

Now, find the sum of these 2k+1 terms:

`S_n=(a_1+a_n)/(2)\cdot n\\ \\S_n=(2k+4k)/(2)\cdot (2k+1)=(6k)/(2)\cdot (2k+1)=3k\cdot (2k+1)=6k^2 +3k`