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Question
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
\[\text { Let the given series be }a_1 + a_2 + a_3 + a_4 + . . . + a_{2n} . \]
\[\text { Now, it is given that }a_1 = 1, a_2 = a a_1 , a_3 = c a_2 , a_4 = a a_3 , a_5 = c a_4 \text { and so on } . \]
\[ \because a_1 = 1\]
\[ \Rightarrow a_1 = 1, a_2 = a, a_3 = ac, a_4 = a^2 c, a_5 = a^2 c^{2,} a_6 = a^3 c^2 , . . . . . \]
\[ \therefore\text { Sum of the 2n terms of the series }, \]
\[ S_n = a_1 + a_2 + a_3 + a_4 + . . . + a_{2n} \]
\[ = 1 + a + ac + a^2 c + a^2 c^2 + . . . + 2n \text { terms }\]
\[ = \left( 1 + a \right) + ac\left( 1 + a \right) + a^2 c^2 \left( 1 + a \right) + . . . + \text { n terms }\]
\[ = \left( 1 + a \right)\left\{ \frac{1 - \left( ac \right)^n}{1 - ac} \right\} \]
\[ = \left( 1 + a \right) \left\{ \frac{\left( ac \right)^n - 1}{ac - 1} \right\}\]
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