MHT CET202120 Sep 2021Evening ShiftMathematicsLimitsActual
If a= _ n 1+2+3+ +n n^2 and b= _ n 1^2+2^2+3^2+ +n^2 n^3 , then
Options
- Aa=b
- B2 a =3 ~b
- Ca=2 b
- D3 a =2 ~b
Correct answer
B. 2 a =3 ~b
Step-by-step solution
aligned & a= _ n 1+2+3+ +n n^2 & = _ n n(n+1) 2(n)(n) = _ n ( 1 2 ) ( n+1 n )= 1 2 _ n (1+ 1 n )= 1 2 & b= _ n 1^2+2^2+3^2+ .+n^2 n^3 & = _ n n(n+1)(2 n+1) 6 n^3 = _ n 2 n^2+3 n+1 6 n^2 & = _ n ( 1 6 ) (2+ 3 n + 1 n^2 )= 2 6 = 1 3 aligned Thus a = 1 2 and b = 1 3 2 a =3 ~b