The n^{th} term of the following series (1 ti | vedclass

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(A) The given series is $(1 	imes 3) + (3 	imes 5) + (5 	imes 7) + (7 	imes 9) + \dots$Let the $n^{th}$ term be $T_n$.The first factors of each te

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$(2n - 1)(2n + 1)$

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(A) The given series is $(1 	imes 3) + (3 	imes 5) + (5 	imes 7) + (7 	imes 9) + \dots$Let the $n^{th}$ term be $T_n$.The first factors of each term are $1, 3, 5, 7, \dots$,which form an Arithmetic Progression $(AP)$ with first term $a = 1$ and common difference $d = 2$.The $n^{th}$ term of this $AP$ is $a_n = a + (n - 1)d = 1 + (n - 1)2 = 1 + 2n - 2 = 2n - 1$.The second factors of each term are $3, 5, 7, 9, \dots$,which form an $AP$ with first term $a = 3$ and common difference $d = 2$.The $n^{th}$ term of this $AP$ is $b_n = a + (n - 1)d = 3 + (n - 1)2 = 3 + 2n - 2 = 2n + 1$.Therefore,the $n^{th}$ term of the series is $T_n = a_n 	imes b_n = (2n - 1)(2n + 1)$.

The $n^{th}$ term of the following series $(1 \times 3) + (3 \times 5) + (5 \times 7) + (7 \times 9) + \dots$ will be

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