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

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

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

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