SOLUTION: use the principals of mathematical induction to prove the following statement 1+5+5^2+...+5^n-1 = 1/4(5n-1) This is calculus

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Question 1104788: use the principals of mathematical induction to prove the following statement 1+5+5^2+...+5^n-1 = 1/4(5n-1)
This is calculus
Answer by math_helper(2461) About Me  (Show Source):
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use the principals of mathematical induction to prove the following statement 1+5+5^2+...+5^n-1 = 1/4(5n-1)
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The right hand side is not correct. It should be +%281%2F4%29%285%5En+-+1%29+
Let +S%5Bn%5D+ = partial sum n (where 5%5E%28n-1%29+ is the highest power in the sum)
Proof by induction:
+S%5B1%5D+=+1+=+%281%2F4%29%285%5E1+-+1%29+
+S%5B2%5D++=+6+=+%281%2F4%29%285%5E2+-+1%29+
Assume +S%5Bk%5D+=+1%2B5%2B5%5E2++ + … + +5%5E%28k-1%29+ = +%281%2F4%29%285%5Ek+-+1%29+ is true for n=k.
Now we need to show that that assumption leads to +S%5Bk%2B1%5D+=+%281%2F4%29%285%5E%28k%2B1%29+-+1+%29+ :
+S%5Bk%2B1%5D+=+1%2B5%2B5%5E2+ + … + +5%5E%28k-1%29+%2B+5%5Ek+
On the right hand side, all but the last term are really just +S%5Bk%5D+
+S%5Bk%2B1%5D+=+S%5Bk%5D+%2B+5%5Ek+
+S%5Bk%2B1%5D+=+%281%2F4%29%285%5Ek+-+1%29+%2B+5%5Ek++
+S%5Bk%2B1%5D+=+%281%2F4%29%285%5Ek+-+1+%2B+4%2A5%5Ek%29+
+S%5Bk%2B1%5D+=+%281%2F4%29%285%2A5%5Ek+-+1%29+
+S%5Bk%2B1%5D+=+%281%2F4%29%285%5E%28k%2B1%29+-+1%29+
DONE.