The following question consist of two statements, one labelled as the 'Assertion (A)' and the other as 'Reason (R)'. You are to examine these two statements carefully and select the answer.
Assertion (A) : `0.3+0.03+0.003+……=(1)/(3).`
Reason (R) : For each (+) ve integer n, let `a_(n)=a+nd`, a and d are real numbers. The, `a_(1)+......+a_(n)=(n)/(2)[2a+(n+1)d].`

Options

A. Both A and R are individually true and R is the correct explanation of A
B. Both A and R are individually true but R is not the correct explanation A
C. A is true but R is false
D. A is false but R is true

Answer

B

Explanation

`(A) :0.3+0.03+0.003+…=(1)/(3)`
Let `S=0.3+0.03+0.003+…`
This is geometric series with
first term, a=0.3 and common ration, `r=(1)/(10).`
So, `S=(a)/(1-r)=(0.3)/(1-(1)/(10))=(0.3xx10)/(9)`
`=(3)/(9)=(1)/(3).` So, (A) is true.
(R) : As given :`a_(n)=a+nd`
`implies" "a_(1)+a_(2)+...+a_(n)`
`=a+d+a+2d+...+a+nd`
`=na+(1+2+...+n)d`
`=na+(n(n+1)d)/(2)=(n)/(2)[2a+(n+1)d]`
So, (R) is also true
Hence, both (A) and (R) are true. but R is not the correct explanation of (A)