Need some help understanding sigma

I'm learning Sigma notation and I think I understand it but something confuses me.

$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$

If I want to find the sum of all integers when n is 15 then this is the same as:

$$\frac{15(15+1)}{2}$$

But, what if i starts at say 4. The formula above wouldn't work because it would include the iterations (incorrect term maybe?) 1,2,3.

So basically is there a formula to get the sum when i isn't 1?

Solution 1:

You can use the above formula twice: if $1\leq k\leq n$, then $$ \sum_{i=k}^ni=\sum_{i=1}^ni-\sum_{i=1}^{k-1}i=\frac{n(n+1)}{2}-\frac{(k-1)k}{2}$$

Solution 2:

Think of $\sum\limits_{i=m}^n = \sum\limits_{i=1}^n - \sum\limits_{i=1}^{m -1} $

That is, cancel out the first $m$ terms of the sequence with another sum so you can still use the formula you have.

Solution 3:

For a generic arithmetic sequence, the sum is $$(\text{number of terms}) \cdot (\text{average of first and last terms}).$$ Notice that your formula has this format too.

The technique that establishes this is usually named after Gauss, and you can read about its derivation on this page under the Sum heading.