Question

Find sum of series

equation is below

Answer 1

\[\sum_{k= 3}^{4} (k^2 - 2k)\]

Answer 2

Well, the formula for the first n-terms of a sequance, starting with n=1 is:
\[\sum_{i=1}^{n}a_i=\left( \frac{ n }{ 2 } \right)(a_1+a_n)\]

I think this is the quickest way to find the value of this sum. So, to find the \(2^{nd}\) and \(4^{th}\) partial sum, and then subtract the \(2^{nd}\) from the \(4^{th}\). By doing this subtraction, I'll be left with the value of the sum of the \(3^{rd}\) through \(4^{th}\) terms. Notice that you're starting at 3 instead of 1. This is why.

The first term is: \(a_1\) = \((1)^2-2(1)\) = -1
Now, plug everything in the formula. You have all that you need
Find it from \(\sum_{i=1}^{3}\) and \(\sum_{i=1}^{4}\)
Follwing this method, you can also do it for larger differences such as from 13 to 57 or so forth.