SOLUTION: Find the sum of series 1+4+13+40+121 to n terms

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Question 1000919: Find the sum of series 1+4+13+40+121 to n terms
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
1%2B4%2B13%2B40%2B121 to n terms


The first thing you should always do is look at the differences between the terms:
4+-+1+=+3
13+-+4+=+9....and so on.
terms: 1, 4, 13, 40, 121, ...
differences: 3, 9, 27, 81
Now we can see a pattern in the differences: 3%5En where n=1,2,....and so on, because 3%5E1+=+3, 3%5E2+=+9, 3%5E3+=+27, 3%5E4+=+81
So the next number must be
121+%2B+3%5E5+=+121+%2B+243+=+364
and the next one is
364+%2B+3%5E6+=+364+%2B+729+=+1093
the nth term can be calculated using the formula:
a%5Bn%5D=%283%5En+-+1%29%2F2
and the sum can be calculated using the formula:
1%2B4%2B13%2B40%2B121+...+%281%2F2%29+%283%5En-1%29
To find the sum of the first n terms of an arithmetic series use the formula,
S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2
S%5Bn%5D=n%281%2B%281%2F2+%29%283%5En-1%29%29%2F2
S%5Bn%5D=n%28%281%2B3%5En%2F2+-1%2F2%29%29%2F2
S%5Bn%5D=%28%28n%2F2%2Bn%2A3%5En%2F2+%29%29%2F2
sum%28%281%2F4+%29%28%283%5En-1%29+n%2Bn%2F2%29%2Cn=1%2Cn%29