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

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Solution

First term $= 1$
Even term $=$ $a$ times the term before it
Odd term $= c$ times the term before it
Let the series be $1, a, a c, a^{2} c, (a c)^{2} . . . . . . . . . . . . . .$
Let $S_{1} = 1 + a c + (a c)^{2} . . . . . n$ terms
$r = a c$
$∴ S_{1} = \frac{1 ((a c)^{n} - 1)}{a c - 1}$
Let $S_{2} = a + a^{2} c + a^{3} c^{2} . . . . . n$ terms $r = a c$ $∴ S_{2} = \frac{a ((a c)^{n} - 1)}{a c - 1}$
$∴$ Total sum $S = S_{1} + S_{2} = \frac{([a c]^{n} - 1) (a + 1)}{a c - 1}$

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

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