How to find the Sum of a geometric sequence in the case where the sum does not start at k=0

Question

Let ($u_n$) be the sequence defined by

$u_0 = −2$ and $∀n ∈ N, u_{n+1} = 5u_n$

2. What is $u_{25} + u_{26} +· · · + u_{35}$ ?

I have been studying geometric sequences in the last couple of days but this is the first time I've come upon a problem like this and I haven't found many helpful tips online in order to solve it. I would extremely appreciate if someone helped me in this :)

Answer 1

Note that $$ \sum_{i = k}^m a_i = \sum_{i = 0}^m a_i - \sum_{i = 0}^{k-1}a_n$$

Answer 2

There are two ways you can look at this.

The first is the more obvious one:

$$u_{25}+\ldots+u_{35}=\left(u_0+\ldots+u_{35}\right)-\left(u_0+\ldots+u_{24}\right)$$

so you just have to compute both sums on the right and subtract.

But really, this is unnecessary. Why? Well, because you can have $u_n$ start on the $25$th term, call it the $0$th term, and your sequence is still a geometric sequence.

To put this more formally, let $v_n=u_{n+25}$. Then $v$ is also a geometric sum with the same ratio and the sum you want is $v_0+\ldots+v_{10}$.

I hope this helps.