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Mathematics Question on Sum of First n Terms of an AP

The sum of the infinite series 1+(1+a)x+(1+a+a2)x2+(1+a+a2+a3)x3+............ 1 + (1 + a) x + (1 + a + a^2) x^2 + (1 + a + a^2 + a^3) x^3+ ............ where 0<a,x<10 < a, x < 1 is

A

1(1−x)(1−a)\frac{1}{(1-x)(1-a)}

B

1(1−x)(1−ax)\frac{1}{(1-x)(1-ax)}

C

1(1−a)(1−ax)\frac{1}{(1-a)(1-ax)}

D

1(1−x)(1+a)\frac{1}{(1-x)(1+a)}

Answer

1(1−x)(1−ax)\frac{1}{(1-x)(1-ax)}

Explanation

Solution

Let S=1+(1+a)x+(1+a+a2)x2+.....∞S= 1+\left(1+a\right)x + \left(1+a+a^{2}\right)x^{2}+.....\infty ∴xS=x+(1+a)x2+.....∞ \therefore xS = x+\left(1+a\right)x^{2} +.....\infty ⇒(1+x)S=1+ax+a2x2+....∞=11−ax\Rightarrow \left(1+x\right)S =1 +ax+a^{2}x^{2}+....\infty= \frac{1}{1-ax} ⇒S=1(1−x)(1−ax)\Rightarrow S= \frac{1}{\left(1-x\right)\left(1-ax\right)}