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Sunday, November 14, 2021

2021/096) Given an arithmetic series a with a common difference d a_1 + a_{2n+1}= z and \sum_{k=1}^{n} a_{2k-1}^2 = x\cdots(1) \sum_{k=1}^{n} a_{2k}^2 = y\cdots(2) Show d = \frac{y-x}{2nz}

We are given

\sum_{k=1}^{n} a_{2k-1}^2 = x\cdots(1)

\sum_{k=1}^{n} a_{2k}^2 = y\cdots(2)

subtract (1) from (2) to get

\sum_{k=1}^{n} (a_{2k}^2- a_{2k-1}^2)  = y-x

Or \sum_{k=1}^{n} (a_{2k}- a_{2k-1})(a_{2k} + a_{2k-1})   = y-x

But (a_{2k}- a_{2k-1}= d common difference so we get

\sum_{k=1}^{n} d(a_{2k} + a_{2k-1})   = y-x

or d \sum_{k=1}^{n} (a_{2k} + a_{2k-1})   = y-x

Or d \sum_{k=1}^{2n} (a_{k})   = y-x\cdots(

as a_k = a_1 + (k-1) d for any k so we have

Now a_k + a_{2n+1-k} = a_1 + (k-1)d + a_1 + (2n+1-k-1)d = 2a_1 + (2n-1) d = = a_1 + a_1 + (2n-1) d = a_1 + a_{2n}

so a_n + a_{n+1}d = a_1 + a_{2n} = z

so a_k + a_{2n+1-k} = z

so 

 d \sum_{k=1}^{2n} (a_{k})

  = d \sum_{k=1}^{n} (a_{k} + a_{2n+1-k})

  = d \sum_{k=1}^{n} z

  = 2dnz

 

  So 2dnz = y-x

    

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