It's bee a long time since I've worked with sums and series, so even simple examples like this one are giving me trouble:
$\sum_{i=4}^N \left(5\right)^i$
Can I get some guidance on series like this? I'm finding different methods online but not sure which to use. I know that starting at a non-zero number also changes things.
My original thought was to do (sum from 0 to N of 5^i) – (sum from 0 to 3 of 5^i) but I'm not sure that's right.
Best Answer
Let $S = a + ar + ar^2 + ar^3 ...$
Then $S-Sr = (a + ar + ar^2 + ar^3 ... ar^n) - (ar + ar^2 + ar^3 + ar^4 ... ar^{n+1}) = a - ar^{n+1}$
Factoring out an S we have $S(1-r) = a-ar^{n+1}$
Finally, $$S = {(a - ar^{n+1})\over(1-r)}$$
In your case, you are trying to find $5^4 + 5^5 + 5^6 ... 5^n$
You can factor out a $5^4$ to get $5^4(1 + 5 + 5^2 ... + 5^{n-4})$
Plugging in corresponding values of $a$ and $r$ into the equation above we have: $$S = 5^4 \times {5^{n-3}-1\over4} $$