6: You should get
If you decided on the add-subtract approach, you should get
If you want to reverse the steps - first try to fix the indexing and only then rewrite the series into a better form - then the first approach would not be possible, since the given series is not geometric and thus the factoring out trick does not work. On the other hand, the second - general - approach still works. Check that you arrive at the same answer in this way.
Do not forget to specify the set of validity of this procedure.
By the way, note that the given series was not a power series, but we summed it up anyway.