Sequence And Series Question 178
Question: The sum of the series $ 1+2x+3x^{2}+4x^{3}+……… $ upto $ n $ terms is
Options:
A) $ \frac{1-(n+1)x^{n}+n{x^{n+1}}}{{{(1-x)}^{2}}} $
B) $ \frac{1-x^{n}}{1-x} $
C) $ {x^{n+1}} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ S_{n} $ be the sum of the given series to $ n $ terms, then $ S_{n}=1+2x+3x^{2}+4x^{3}+……..+n{x^{n-1}} $ ?..(i) $ xS_{n}=\text{ }x+2x^{2}+3x^{2}+………..+nx^{n} $ ?..(ii) Subtracting (ii) from (i), we get $ (1-x)S_{n}=1+x+x^{2}+x^{3}+…..to $ $ n $ terms $ -nx^{n} $ $ =( \frac{(1-x^{n})}{(1-x)} )-nx^{n} $
$ \Rightarrow S_{n}=\frac{(1-x^{n})-nx^{n}(1-x)}{{{(1-x)}^{2}}}=\frac{1-(n+1)x^{n}+n{x^{n+1}}}{{{(1-x)}^{2}}} $ .
BETA
