Prove that for all sequences $a_k, b_k$ that for all $n \in \mathbb{N}$: $\sum^n_{k=1}(a_k+b_k) = \sum^n_{k=1}a_k + \sum^n_{k=1} b_k$
So it sounds like induction but what confuses me are the series here, should I just assume two arbitrary sequences and do a proof by induction? But how do I do that with two arbitrary sequences?
On
You certainly can do this by induction.
When n= 1, this is just $(a_1+ b_1)= (a_1)+ (b_1)$ which is true because the parentheses don't mean anything- I just put them in so you could see how the series were written.
Now suppose that $\sum_{k=1}^n (a_k+ b_k)= \sum_{k= 1}^n a_k+ \sum_{k= 1}^n b_k$. Then $\sum_{k= 1}^{n+1} a_k+ b_k= \sum_{k= 1}^n a_k+ b_k+ a_{n+1}+ b_{n+1}= \left(\sum_{k= 1}^n a_k+ a_{n+1}\right)+ \left(\sum_{k=1}^n b_k+ b_{n+1}\right)= \sum_{k=1}^{n+1} a_k+ \sum_{k=1}^{n+1} b_k$
All you need is the commutative property of addition (+). Then, \begin{align} \sum_k a_k + b_k & = a_1+b_1+\dots+a_n+b_n \\ & =a_1+\dots + a_n + b_1 + \dots + b_n \\ & = \sum_k a_k + \sum_k b_k. \end{align}