Find the sum of 2n terms of the series whose every even term is 'a'

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.

Solution:

Let the given series be $a_{1}+a_{2}+a_{3}+a_{4}+\ldots+a_{2 n}$.

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.

$\because a_{1}=1$

$\Rightarrow a_{1}=1, a_{2}=a, a_{3}=a c, a_{4}=a^{2} c, a_{5}=a^{2} c^{2,} a_{6}=a^{3} c^{2}, \ldots$

$\therefore$ Sum of the $2 n$ terms of the series,

$S_{n}=a_{1}+a_{2}+a_{3}+a_{4}+\ldots+a_{2 n}$

$=1+a+a c+a^{2} c+a^{2} c^{2}+\ldots+2 n$ terms

$=(1+a)+a c(1+a)+a^{2} c^{2}(1+a)+\ldots+n$ terms

$=(1+a)\left\{\frac{1-(a c)^{n}}{1-a c}\right\}$

$=(1+a)\left\{\frac{(a c)^{n}-1}{a c-1}\right\}$