SOLUTION: For the sequence 1/3, 1/3^2, 1/3^3, 1/3^4, 1/3^5,..., find its fifth partial sum and its sixth partial sum. Help would be appreciated, thanks!

Algebra ->  Sequences-and-series -> SOLUTION: For the sequence 1/3, 1/3^2, 1/3^3, 1/3^4, 1/3^5,..., find its fifth partial sum and its sixth partial sum. Help would be appreciated, thanks!      Log On


   

Question 933251: For the sequence 1/3, 1/3^2, 1/3^3, 1/3^4, 1/3^5,..., find its fifth partial sum and its sixth partial sum.
Help would be appreciated, thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Each term of 
1/3, 1/3^2, 1/3^3, 1/3^4, 1/3^5, ...
is in the form a%2A%28r%29%5E%28n-1%29 where a+=+1%2F3 and r+=+1%2F3


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S%5Bn%5D+=+a%2A%28%281-r%5En%29%2F%281-r%29%29 Geometric series nth partial sum formula.


S%5Bn%5D+=+%281%2F3%29%2A%28%281-%281%2F3%29%5En%29%2F%281-1%2F3%29%29 Plug in r+=+1%2F3


S%5Bn%5D+=+%281%2F3%29%2A%28%281-%281%2F3%29%5En%29%2F%282%2F3%29%29


S%5Bn%5D+=+%281%2F3%29%2A%283%2F2%29%2A%281-%281%2F3%29%5En%29


S%5Bn%5D+=+%281%2F2%29%2A%281-%281%2F3%29%5En%29


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Fifth Partial Sum (n = 5)


S%5Bn%5D+=+%281%2F2%29%2A%281-%281%2F3%29%5En%29


S%5B5%5D+=+%281%2F2%29%2A%281-%281%2F3%29%5E5%29


S%5B5%5D+=+%281%2F2%29%2A%281-1%2F243%29


S%5B5%5D+=+%281%2F2%29%2A%28242%2F243%29


S%5B5%5D+=+121%2F243


The fifth partial sum is 121%2F243 which is approximately 0.49794238683128%0D%0A%0D%0A%0D%0A%0D%0A (I'd stick to the exact answer whenever you can)


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Sixth Partial Sum (n = 6)


S%5Bn%5D+=+%281%2F2%29%2A%281-%281%2F3%29%5En%29


S%5B6%5D+=+%281%2F2%29%2A%281-%281%2F3%29%5E6%29


S%5B6%5D+=+%281%2F2%29%2A%281-1%2F729%29


S%5B6%5D+=+%281%2F2%29%2A%28728%2F729%29


S%5B6%5D+=+364%2F729


The sixth partial sum is exactly 364%2F729 which is approximately 0.49931412894376


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