Given a sequence of real number $a_{1}$,$a_{2}$,...,$a_{n+1}$ show that $\sum_{k=1}^{n} (a_{k+1}-a_{k}) = a_{n+1}-a_1$
I am stuck on this problem we have been given by my lecturer. I don't have much experience with sums, but I have tried searching my way to an answer, but I end up with nothing. The textbook for the course doesn't really help and I could use a pointing finger to start solving the problem.
I have tried constructing the sum's sigma notation from just the given sequence and tried taking things from there, but I do not get any further than the construction and maybe(?) reversing the index shift (if that is even allowed):
$\sum_{k=1}^{n+1} a_k$ = $\sum_{k=0}^{n} a_{k+1}$
Thanks in advance :)
Edit:
I now realize the pattern of $\sum_{k=1}^{n} (a_{k+1}-a_{k})$. But I do not see why $a_1$ is subtracted from $a_{n+1}$. Induction is not a topic that my lecturer has covered yet either... I know however where to take it from now, thanks for the fast responses :)