The sum of $2 n$ terms of a series of which every even term is $^{'} a^{'}$ times the terms before it, and every odd term $^{'} c^{'}$ times the terms before it, the first term being unity, is

\frac{(1 - a) (a^{n} c^{n} - 1)}{a c - 1}

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\frac{(1 + a) (a^{n} c^{n} - 1)}{a c + 1}

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\frac{(1 + a) (a^{n} c^{n} - 1)}{a c - 1}

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Solution

The correct option is C $\frac{(1 + a) (a^{n} c^{n} - 1)}{a c - 1}$
$\begin{matrix} T_{1} = 1 T_{2} = a T_{3} = C a T_{4} = C a^{2} T_{2 n} = a \frac{2 n}{2} \cdot C \frac{2 n}{2} - 1 = a^{n} C^{n - 1} 5_{2 n} = 1 + a + c a . . . . . a^{n} C^{n - 1} = 1 + [a + c a^{2} + c^{2} a^{3} . . . a^{n} c^{n - 1}] + [c a + c^{2} a^{2} + c^{3} a^{3} . . . . . c^{n - 1} a^{n - 1}] = 1 + \frac{a (a^{n} c^{n - 1})}{a c - 1} + \frac{a c (a^{n - 1} c^{n - 1} - 1)}{a c - 1} = \frac{(a^{n} c^{n} - 1) (a + 1)}{a c - 1} \end{matrix}$

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The sum of $2 n$ terms of a series of which every even term is $^{'} a^{'}$ times the terms before it, and every odd term $^{'} c^{'}$ times the terms before it, the first term being unity, is