Question

Which of the following can be terms (not necessarily consecutive) of any A.P.? a. 1,6,19 b. `sqrt(2),sqrt(50),sqrt(98)` c. `log2,log16 ,log128` d. `sqrt(2),sqrt(3),sqrt(7)`
A. 1,6,19
B. `sqrt(2).sqrt(50),sqrt(98)`
C. log 2,log 16 , log128
D. `sqrt(2),sqrt(3),sqrt(7)`

Answer 1

Correct Answer - A::B::C
Let a,b,c be pth,qth and rth terms of A.P. Then
a=A+(p-1)D,b=A+(q-1)D,c=A+(r-1)D
`rArr(r-q)/(q-p)=(c-b)/(b-a)` is rational number.
now for 1,6,19,`(r-q)/(q-p)=(19-6)/(6-1)` is a rational number.
For `sqrt2,sqrt(50),sqrt(98),(r-q)/(q-p)=(sqrt(98)-sqrt(50))/(sqrt(50)-sqrt2)=(7sqrt2-5sqrt2)/(5sqrt2-sqrt2)`
`=1/2` is a rational number.
For log2, log 16, log 128
`(r-q)/(q-p)=(log128-log16)/(log16-log2)=(7log2-4log2)/(4log2-log2)=1`
is a rational number.
But for `sqrt2,sqrt3,sqrt7,(r-q)/(q-p)` is not a rational number.