Question

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

Answer 1

Let the series be $a_1+a_2+a_3+....+a_{2n}$ It is given that $a_1=1,a_2=a,a_3=ac,a_4=a^{2}c,a_5=a^2{c}^2,$......

$\therefore$ Sum of 2n term

$a_1+a_2+a_3+....+a_{2n}$

$=1+a+ac+a^{2}c+a^2{c}^2+....+2n$ term

$=(1+a)+ac(1+a)+a^2{c}^2(1+a)+....+n$ term

$=(1+a)\frac{(1-(ac{)}^n)}{1-ac}$

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