Question

Consider an AP with a as the first term and d is the common difference such that `S_(n)` denotes the sum to n terms and `a_(n)` denotes the nth term of the AP. Given that for some m,`n inN,(S_(m))/(S_(n))=(m^(2))/(n^(2))(nen)`.
Statement 1 `d=2a` because
Statement 2 `(a_(m))/(a_(n))=(2m+1)/(2n+1)`.
A. Statement 1 is true, Statement 2 is true, Statement 2 is a corrct explanation for Statement 1.
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
C. Statement 1 is true, Statement 2 is false.
D. Statement 1 is false, Statement 2 is true.

Answer 1

Correct Answer - C
`:.(S_(m))/(S_(n))=(m^(2))/(n^(2))`
Let `S_(m)=m^(2)k,S_(n)=n^(2)k`
`:.a_(m)=S_m-S_(m-1)=m^(2)k-(m-1)^(2)k`
`a_(m)=(2m-1)k`
Sililarly, `a_(n)=(2n-1)k:.(a_(m))/(a_(n))=(2m-1)/(2n-1)`
Statement 1 is false.
Also, `:. a_(1)=k,a_(2)=3k,a_(3)=5k,"...."`
Given, `a_(1)=a=k`
`:.a_(1)=a,a_(2)=3a,a_(3)=5a,"...."`
`:.` Common difference `d=a_(2)-a_(1)=a_(3)-a_(2)="...."`
`implies d=2a`
`:.` Statement 1 is true.