Find the sum of 2n terms of the series whose every even term is 'a' times the term before it and every odd term is 'c' times the term before it, the first term being unity.

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Solution

$Let the given series be a_{1} + a_{2} + a_{3} + a_{4} + . . . + a_{2 n} . Now, it is given that a_{1} = 1, a_{2} = a a_{1}, a_{3} = c a_{2}, a_{4} = a a_{3}, a_{5} = c a_{4} and so on . ∵ a_{1} = 1 \Rightarrow a_{1} = 1, a_{2} = a, a_{3} = a c, a_{4} = a^{2} c, a_{5} = a^{2} c^{2,} a_{6} = a^{3} c^{2}, . . . . . ∴ Sum of the 2 n terms of the series, S_{n} = a_{1} + a_{2} + a_{3} + a_{4} + . . . + a_{2 n} = 1 + a + a c + a^{2} c + a^{2} c^{2} + . . . + 2 n terms = (1 + a) + a c (1 + a) + a^{2} c^{2} (1 + a) + . . . + n terms = (1 + a) \{\frac{1 - {(a c)}^{n}}{1 - a c}\} = (1 + a) \{\frac{{(a c)}^{n} - 1}{a c - 1}\}$

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