Question

The sum of the series $1\cdot 3^2+2\cdot 5^2+3\cdot 7^2+...$ upto $20$ terms is

• A. $188090$
• B. $189080$
• C. $199080$
• D. None of these

Answer 1

The correct option is A $188090$

Here, $T_n$ of the AP $1,2,3,...=n$
And $T_n$ of the AP $3,5,7,...=2n+1$
Therefore, $T_n$ of given series is
$T_n=n{(2n+1)}^2=4n^3+4n^2+n$
Hence, $S=\sum _1^{20}{T}_n$ $=4{\sum }_1^{20}{n}^3+4{\sum }_1^{20}{n}^2+{\sum }_1^{20}n$

$=4\times \frac{1}{4}\times {20}^2\times {21}^2+4\times \frac{1}{6}\times 20\times 21\times 41+\frac{1}{6}\times 20\times 21=188090$