Sequence And Series Question 93
Question: The sum of the series $ 1+3x+6x^{2}+10x^{3}+……..\infty $ will be
Options:
A) $ \frac{1}{{{(1-x)}^{2}}} $
B) $ \frac{1}{1-x} $
C) $ \frac{1}{{{(1+x)}^{2}}} $
D) $ \frac{1}{{{(1-x)}^{3}}} $
Show Answer
Answer:
Correct Answer: D
Solution:
Let $ S=1+3x+6x^{2}+10x^{3}+…..\infty $
$ \Rightarrow $ $ x.S=x+3x^{2}+6x^{3}+…….\infty $ Subtracting $ S(1-x)=1+2x+3x^{2}+4x^{3}+…….\infty $
$ \Rightarrow $ $ x(1-x)S=x+2x^{2}+3x^{3}+…….\infty $ Again subtracting,
$ \Rightarrow $ $ S[(1-x)-x(1-x)]=1+x+x^{2}+x^{3}+……..\infty $
$ \Rightarrow $ $ S[(1-x)(1-x)]=\frac{1}{1-x}\Rightarrow S=\frac{1}{{{(1-x)}^{3}}} $
BETA
