How to proof linearity property of summations with induction

We proceed as normal with induction (see Guido Kanschat's post for the general process). When $n=1$, we have $\sum_{k=1}^1 \left(ca_k+b_k\right)=ca_1+b_1=c\sum_{k=1}^1 a_k+\sum_{k=1}^1 b^k$, so our base case holds. Now suppose that for some $k\in\Bbb{N}$ our proposition holds (that is $\sum_{i=1}^k \left(ca_i+b_i\right) = c\sum_{i=1}^k a_i+\sum_{i=1}^k b_i$).

Now we have

\begin{eqnarray} \sum_{i=1}^{k+1} \left(ca_i+b_i\right) &=& \sum_{i=1}^k \left(ca_i+b_i\right)+(ca_{k+1}+b_{k+1})\\ &=&\left(c\sum_{i=1}^k a_i+ca_{k+1}\right)+\left(\sum_{i=1}^k b_i+b_{k+1}\right)\\ &=& c\left(\sum_{i=1}^k a_i+a_{k+1}\right)+\left(\sum_{i=1}^k b_i+b_{k+1}\right)\\ &=&c\sum_{i=1}^{k+1}a_i+\sum_{i=1}^{k+1}b_i. \end{eqnarray}

I skipped a few steps (hence the ellipsis) in this last equation, I hope you take the time to figure out why the equality holds.