Sequence And Series Question 397
Question: What is the sum of the series 0.5 + 0.55 + 0.555 +… to n terms?
Options:
A) $ \frac{5}{9}[ n-\frac{2}{9}( 1-\frac{1}{10^{n}} ) ] $
B) $ \frac{1}{9}[ 5-\frac{2}{9}( 1-\frac{1}{10^{n}} ) ] $
C) $ \frac{1}{9}[ n-\frac{5}{9}( 1-\frac{1}{10^{n}} ) ] $
D) $ \frac{5}{9}[ n-\frac{1}{9}( 1-\frac{1}{10^{n}} ) ] $
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Given $ 0.5+0.55+0.555+…..ton $ $ =5[0.1+0.11+0.111+…….tonterms] $ $ =\frac{5}{9}[0.9+0.99+0.999+…….tonterms] $ $ =\frac{5}{9}[ \frac{9}{10}+\frac{99}{100}+\frac{999}{1000}+……to,n,terms ] $ $ =\frac{5}{9}[ \begin{aligned} & ( 1-\frac{1}{10} )+( 1-\frac{1}{100} )+( 1-\frac{1}{1000} )+…. \\ & ,tonterms \\ \end{aligned} ] $ $ =\frac{5}{9}[ \begin{aligned} & ( 1-\frac{1}{10} )+( 1-\frac{1}{10^{2}} )+( 1-\frac{1}{10^{3}} )+…. \\ & ( 1-\frac{1}{10^{n}} ) \\ \end{aligned} ] $ $ =\frac{5}{9}[ n-( \frac{1}{10}+\frac{1}{10^{2}}+….\frac{1}{10^{n}} ) ] $ $ =\frac{5}{9}[ n-\frac{1}{10}\frac{{ 1-{{( \frac{1}{10} )}^{n}} }}{( 1-\frac{1}{10} )} ] $ $ =\frac{5}{9}[ n-\frac{1}{9}( 1-\frac{1}{10^{n}} ) ] $
BETA
