Find the nth Partial Sum Sn of the series sum 1(2kappa −1)(2kappa +1) kindly solve fast.

Question
Series
asked 2021-02-25
Find the nth Partial Sum Sn of the series
\(\displaystyle\sum{1}{\left({2}\kappa−{1}\right)}{\left({2}\kappa+{1}\right)}\)
kindly solve fast.

Answers (1)

2021-02-26
\(\displaystyle{\sum_{{{k}={1}}}^{{{k}={n}}}}{1}{\left[{\left({2}{k}−{1}\right)}{\left({2}{k}+{1}\right)}\right]}\)
\(\displaystyle{\sum_{{{k}={1}}}^{{{k}={n}}}}{1}{\left[{4}{\left({k}^{{2}}\right)}−{1}\right]}\)
\(\displaystyle{\sum_{{{k}={1}}}^{{{k}={n}}}}{\left[{4}{\left({k}^{{2}}\right)}\right]}-∑{\left[{1}\right]}\)
\(\displaystyle{4}{\sum_{{{k}={1}}}^{{{k}={n}}}}{\left[{k}^{{2}}\right]}-∑{\left[{1}\right]}\)
\(\displaystyle{4}{\left({1}^{{{2}}}\right)}+{2}^{{{2}}}+{3}^{{{2}}}+\ldots+{n}^{{{2}}}{)}-{\left({1}+{2}+{3}+\ldots+{n}\right)}\)
Sum of square of natural number =\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}{\left({2}{n}+{1}\right)}}}{{{6}}}}\)
Sum of natural number =\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}}}{{{2}}}}\)
\(\displaystyle{\frac{{{4}{n}{\left({n}+{1}\right)}{\left({2}{n}+{1}\right)}}}{{{6}}}}-{\frac{{{n}{\left({n}+{1}\right)}}}{{{2}}}}\)
\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}}}{{{2}}}}\times{\left[{\frac{{{4}{\left({2}{n}+{1}\right)}}}{{{3}}}}-{1}\right]}\)
\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}}}{{{2}}}}{\left[{\frac{{{8}{n}+{4}-{3}}}{{{3}}}}\right]}\)
\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}}}{{{2}}}}{\left[{\frac{{{8}{n}+{1}}}{{{3}}}}\right]}\)
\(\displaystyle{\frac{{{n}{\left({n}+{1}\right)}{\left({8}{n}+{1}\right)}}}{{{6}}}}\)
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