Find the $20^{t h}$ term of the series $2 \times 4 + 4 \times 6 + 6 \times 8 + . . . . + n$ term.

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Solution

The series is $2 \times 4 + 4 \times 6 + 6 \times 8 + . . . . . . . . . . + n$ terms
we will break into two parts like as $\to 2, 4, 6...... & 4, 6, 8......$
find $n^{t h}$ terms separately of given series $\to$
$\begin{matrix} T_{n} = {n^{t h} t e r m s o f 2, 4, 6....} {n^{t h} t e r m s o f 4, 6, 8.....} T_{n} = {2 + (n - 1) 2} {4 + (n - 1) 2} & [I n I^{s t} t e r m s a = 2, d = 2] T_{n} = {2 + 2 n - 2} {4 + 2 n - 2} & [I n 2^{n d} t e r m s a = 4, d = 2] T_{n} = 2 n (2 + 2 n) & [U s i n g a_{n} = a + (n - 1) d] P u t n = 20 T_{20} = 2 \times 20 (2 + 2 \times 20) T_{20} = 40 (2 + 40) T_{20} = 40 (42) T_{20} = 1680 \end{matrix}$

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