Do summations always increment by one?
Having a more of background in programming than math. I am just learning about summations and I look at them as loops that increment by one.
If my assumption correct what an equivalent method / form to increment by values not equal to one?
Do summations always increment by one?
That depends on what you mean by summation.
E.g $$ S = \sum_{i = 1}^n s_i = \sum_{i \in I} s_i $$ means a summation, where the indices are picked from $I = \{1,\ldots, n\}$. The index set $I$ could be traversed by starting at $1$ and incrementing to $n$ in steps of $+1$. But that is just one way of $n!$ possible ways to sum the $s_i$ up.
Having a more of background in programming than math. I am just learning about summations and I look at them as loops that increment by one.
If my assumption correct what an equivalent method / form to increment by values not equal to one?
You could start with $n$ and finish with $1$. Or you first use the odd indices ascending and the the even ones descending. $$ S = \sum_{\overset{i \in I}{ i \bmod 2 = 1}} s_i + \sum_{\overset{i \in I}{ i \bmod 2 = 0}} s_i $$
It should lead to the same sum, as order does not matter to the finite index set.
No, summations do not always increment by one. In fact, summations can be over many kinds of sets, possibly not even restricted to integers or countable cardinality. Some examples from Wikipedia:
An example of a proof involving a summation over an uncountable set can be seen here on SE: The sum of an uncountable number of positive numbers
This may at first seem like a problem, but once you start thinking about how to solve it the solution becomes obvious and attests to the beauty of simplicity in math.
Let's solve the problem of being able to increment by any real number: simply multiply $i$ by $p$, the desired real number.
Using negative numbers? Increment by a negative number, etc.
Has anyone considered using programming notation? For example for an increment by 3
$$\sum_{\substack{n=0 \\ n+=3}}^{\inf} n$$
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. If it were understood to be the += operation, you could even just put '3' in the center and have it understood that this means increment.
You could even make stranger increment conditions like in terms of remainders if this is what you want
$$\sum_{\substack{n=0 \\ n\%3=1}}^{\inf} n$$
would give every number just after a multiple of 3. Admittedly for this specific case you could just start at 1 and index by 3 but this notation could be extended to odder situations.
In programming, many languages provide for an optional step in the loop. It defaults to $1$, but you can set it to something else. In math, it is similar. The default is to increment by $1$. If you want to add up all the multiples of $3$ from $3$ to $30$, you could write $$\sum_{i=1}^{10}3i$$, where $i$ increments by $1$ and the factor $3$ gets you the increment you really want. In programming you can do the same thing. You can write for i=1 to 10;k=3*i and use k instead of i for your computations. You can also write for i=3 to 30 step 3. In math you can write $$\sum_{ \substack {i=3\\ i\equiv 0 \pmod 3}}^{30}i$$ to get a step of 3. It seems fairly rare in math, more common in programming, in my experience
