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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