Summation Skip Notation

Question

The following notation means to sum 1 to N:

$$\sum_{n=1}^N n$$

Is there a notation to not increment by one for each step, but, say, 10?

Answer 1

As long as you have a constant skip size, you can handle it by multiplying the subscript by the skip size wherever it appears inside the summation: in your example you’d get

$$\sum_{n=1}^N10n\;.$$

Of course that gives you the sum of all $N$ values from $10$ through $10N$. If you wanted just the sum of those multiples of $10$ that are no bigger than $N$, you’d have to adjust the upper limit:

$$\sum_{n=1}^{\lfloor N/10\rfloor}10n\;.$$

If you wanted to start with $n=1$ and increase in steps of $10$, for a total of $N$ terms, it would be

$$\sum_{n=0}^{N-1}(1+10n)\;.$$

And if you wanted to start with $n=1$ and increase in steps of $10$ up through a maximum value of $N$, it would be

$$\sum_{n=0}^{\left\lfloor\frac{N-1}{10}\right\rfloor}(1+10n)\;.$$

Answer 2

You could just write: $$ \sum_{k = 1}^{N/10} 10 k $$ Another option is to write conditions under the sum sign: $$ \sum_{\substack{1 \le n \le N \\ 10 \mid n}} n $$

Answer 3

Has anyone considered using programming notation? For example for an increment by 3

$$\sum_{\substack{n=0 \\ n+=3}}^{\inf} x$$

I think even better would be to put the increment in the middle of the sum (start under, increment middle, end top), but latex lacks support of that without making it look like the contents of the sum.