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  4. If displaystyle∑r=1n r4=I(n) then displaystyle∑r=1n(2 r-1)4 is equal to

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Q. If $ \displaystyle\sum_{r=1}^{n} r^{4}=I(n)$ then $\displaystyle\sum_{r=1}^{n}(2 r-1)^{4}$ is equal to

Sequences and Series

Solution:

$ I(2 n)=1^{4}+2^{4}+3^{4}+\ldots+(2 n-1)^{4}+(2 n)^{4} $
$=\left(1^{4}+3^{4}+5^{4}+\ldots+(2 n-1)^{4}\right) +2^{4}\left(1^{4}+2^{4}+3^{4}+4^{4}+\ldots n^{4}\right) $
$= \displaystyle\sum_{r=1}^{n}(2 r-1)^{4}+16 \cdot I(n)$
$\Rightarrow \displaystyle\sum_{r=1}^{n}(2 r-1)^{4}=I(2 n)-16 I(n)$