Sigma Notation For Indexing Over Non-Consecutive Integers

Question

My Question: How do I know by which number to increase my adding from start to finish with Sigma As far as I have learned when using sigma with a rule from start and finish your number always increases by one. (I had a like situation with square roots until I learned there was a markup for things like 3 square and 4 square etc...)

So for the above equation assume the start is 2 and the finish is 10. Is there any way to tell someone to increase by say two when doing this equation instead of one.

Answer 1

There is no special notation needed for that. Use a summation $\displaystyle\sum_{i=1}^5(\ldots)$ and write $2i$ everywhere you want your quantity that increases from $2$ to $10$ in steps of $2$.

Answer 2

@Sal's answer is good. Another way, which avoids changing the expression in the sum, is

$$\sum_{2\le i\le 10,\ i\text{ even}}4i$$

or perhaps

$$\sum_{2\le i\le 10,\ 2\mid i}4i$$

If you think it is needed, you could add a statement that $i$ is an integer, but that does not seem to be necessary.

Answer 3

I usually use the generalised sigma notation to represent indexing over non-consecutive integers:

$$\sum\limits_{\underset{\text{k is even}}{2 \leq k \leq 10} } 4k$$

In some cases, the generalised form is much less cumbersome than the delimited form. For example, the sum of all odd positive integers below 100 is written in the generalised form as follows:

$$\sum\limits_{\underset{\text{k is odd}}{1 \leq k \leq 100} } k^2$$

In the delimited form this would be written as:

$$\sum\limits_{k = 0}^{49} (2k + 1)^2$$

Personally, I think the reader can easily understand the intent of the summation from the generalised form compared to the delimited form.