SOLUTION: Find the sum of n term of the series. 1+4+13+40+........ .

Question 1089104: Find the sum of n term of the series.
1+4+13+40+........ .
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
If you calculate the differences between consecutive terms,
you find that those differences form a geometric sequence
(or geometric progression, or GP, if those are the preferred names where you live).
The terms are
,
,
,
.
So,
The differences form a geometric sequence/progression with common ratio r=3.
So, you can calculate each term as the sum of that geometric sequence/progression.
You can also figure it out from polynomial products,
knowing that ,
and for you have
.
Either way, you end up with

Each term is the sum of all the terms before,
The sum of you have to calculate is

.
The sum in the formula is related to the one we had calculated before:
, so

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