SOLUTION: What is the formula for the sequence 5,5,35,65,275. . .

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Question 1014011: What is the formula for the sequence 5,5,35,65,275. . .
Answer by Edwin McCravy(20066) About Me  (Show Source):
You can put this solution on YOUR website!

First we'll try a sum of nth powers with the first two terms:
If that doesn't work, we'll try the difference of nth powers
a%5Bn%5D%22%22=%22%22x%5En%2By%5En
5%22%22=%22%22x%5E1%2By%5E1
5%22%22=%22%22x%5E2%2By%5E2
system%28x%2By=5%2Cx%5E2%2By%5E2=5%29

No good. Comes out with imaginary values for x and y
So we try a difference of powers for the first two terms:
a%5Bn%5D%22%22=%22%22x%5En-y%5En
5%22%22=%22%22x%5E1-y%5E1
5%22%22=%22%22x%5E2-y%5E2
system%28x-y=5%2Cx%5E2-y%5E2=5%29

matrix%283%2C1%2C%0D%0Ax-5=y%2C%0D%0A3-5=y%2C%0D%0A-2=y%29
So the formula we get from the first two terms,
substituting x=3 and y=-2 in
a%5Bn%5D%22%22=%22%22x%5En-y%5En
becomes
a%5Bn%5D%22%22=%22%223%5En-%28-2%29%5En
But we must check to see if it holds for all the given
numbers in the series:
a%5B1%5D%22%22=%22%223%5E1-%28-2%29%5E1%22%22=%22%223-%28-2%29%22%22=%22%223%2B2%22%22=%22%225
a%5B2%5D%22%22=%22%223%5E2-%28-2%29%5E2%22%22=%22%229-%284%29%22%22=%22%225
a%5B3%5D%22%22=%22%223%5E3-%28-2%29%5E3%22%22=%22%2227-%28-8%29%22%22=%22%2227%2B8%22%22=%22%2235
a%5B4%5D%22%22=%22%223%5E4-%28-2%29%5E4%22%22=%22%2281-%2816%29%22%22=%22%2265
a%5B5%5D%22%22=%22%223%5E5-%28-2%29%5E5%22%22=%22%22243-%28-32%29%22%22=%22%22243%2B32%22%22=%22%22275
So it works for all the given terms, so the formula is:
a%5Bn%5D%22%22=%22%223%5En-%28-2%29%5En
Edwin