SOLUTION: If there are (2n+1) terms in an arithmetic series, prove that the ratio of the sum of odd place terms to the sum of even place terms is {{{ n^4-1 }}} : {{{ n^4-n^3+n^2-n }}} .
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Question 569522: If there are (2n+1) terms in an arithmetic series, prove that the ratio of the sum of odd place terms to the sum of even place terms is : .
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Suppose our sequence is a, a+d, a+2d, ..., a+2nd (this ensures that there are 2n+1 terms). Denote S_o and S_e to be the sum of the odd place terms and even place terms respectively. Then,
 + (a+4d) + ... + (a+2nd))
 + (a+3d) + ... + (a+(2n-1)d))
S_o includes n+1 terms and S_e includes n terms. By equating the coefficients of a and d in each series, we have
 + n(n+1)d)
Therefore,  + n(n+1)d}{an + n^2d} = \frac{(n+1)(a+nd)}{n(a+nd)} = \frac{n+1}{n})
. Multiplying both numerator and denominator by n^3 - n^2 + n - 1 yields the desired result.
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