nth term of the series 1 + frac{3}{7} + frac{ | vedclass

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(B) The given series is $1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots$ The numerators are $1, 3, 5, 7, \dots$,which form an Arithmetic Prog

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$\frac{2n-1}{7^{n-1}}$

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(B) The given series is $1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots$ The numerators 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 $T_n(AP) = a + (n-1)d = 1 + (n-1)2 = 2n - 1$. The denominators are $7^0, 7^1, 7^2, 7^3, \dots$,which form a Geometric Progression $(GP)$ with first term $a = 1$ and common ratio $r = 7$. The $n$th term of this $GP$ is $T_n(GP) = ar^{n-1} = 1 	imes 7^{n-1} = 7^{n-1}$. Therefore,the $n$th term of the given series is $T_n = \frac{2n-1}{7^{n-1}}$.

$n$th term of the series $1 + \frac{3}{7} + \frac{5}{7^2} + \frac{7}{7^3} + \dots$ is

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