How to Solve Summation Notation

Question

I am having trouble with this equation in a class I am taking and I am trying to understand it: $$\sum_{i=47}^{i=136} M_i$$

We have to solve for the problem below but the hint our professor gave us was subtraction. I am confused because it is my understanding that the bottom was the starting point and the top was the ending point. So wouldn't $M_i$ start with $M_{47}$ and work up from there? Any help is much appreciated.

Answer 1

You are right, the summation you are given is over the sequence 47, 48, 49, $\dots$, 136.

You are probably used to summation formulas that start with 1. So can you rewrite the given summation in terms of those familiar ones? $$ \sum_{i=47}^{136} M_i = \sum_{i=1}^{?} M_i - \sum_{i=1}^{?} M_i $$

Answer 2

The summation can be rewritten as:

$$\sum_{i=47}^{i=136} M_i =\sum_{i=0}^{i=136} M_i - \sum_{i=0}^{i=46} M_i$$

The reason behind this is that the when summing from $i=0$ to $i=136$ we would be also including all $M_i$ from $i=0$ to $i=46$. We can then use subtract the sum of all numbers in that range to get the final result.

Answer 3

Without knowing what $M_i$ is, I can only guess that he might be suggesting that you do $$\sum_{i=47}^{i=136} M_i = \sum _{i=0}^{i=136} M_i - \sum_{i=0}^{i=46}M_i $$