SOLUTION: Simplify {{{ (1/3)+(3/(3^2))+((3^2)/(3^3))+((3^3)/(3^4))+... }}} + {{{ ((3^2999)/(3^3000)) }}}

Algebra ->  Exponents -> SOLUTION: Simplify {{{ (1/3)+(3/(3^2))+((3^2)/(3^3))+((3^3)/(3^4))+... }}} + {{{ ((3^2999)/(3^3000)) }}}      Log On


   

Question 1104110: Simplify + +%28%283%5E2999%29%2F%283%5E3000%29%29+
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!


+S%5B5%5D+=+4%2F3+%2B+3%5E4%2F3%5E5+=+4%2A3%5E4%2F3%5E5+%2B+3%5E4%2F3%5E5+=+5%2F3+
: : :
+S%5Bn%5D+=+n%2F3+ <— Assertion
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Proof of +S%5Bn%5D+=+n%2F3+ by induction…
We already see it is true for n=3,4,5, assume it is true for n.
Then


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+S%5B3000%5D+=+3000%2F3+=+highlight%281000%29+
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To the student:
+S%5Bn%5D+ = partial sum (S is often used for this purpose). I found the first few partial sums to see if there was a pattern. That allowed me to assert S%5Bn%5D+=+n%2F3, then I proved that assertion by induction so I could safely use it to compute +S%5B3000%5D+.