Question

Let `f(n)` denote the `n^(th)` terms of the seqence of `3,6,11,18,27,….` and `g(n)` denote the `n^(th)` terms of the seqence of `3,7,13,21,….` Let `F(n)` and `G(n)` denote the sum of `n` terms of the above sequences, respectively. Now answer the following:
`lim_(ntooo)(f(n))/(g(n))=`
A. `0`
B. `1`
C. `2`
D. `oo`

Answer 1

Correct Answer - B
`(b)` `S=3+6+11+18+...+t_(n)`
`S=3+6+11+...+t_(n-1)+t_(n)`
`thereforeoverline(0=3+(3+5+7+9+...(n-1)"terms"-t_(n))`
`:.t_(n)=n^(2)+2`
Similarly `nth` terms of `g(n)=n^(n)+n+1`
`:.lim_(ntooo)(n^(2)+2)/(n^(2)+n+1)=1`
`F(n)=sum(n^(2)+2)`
`=(n(n+1)(2n+1))/(6)+2n`
`=(n(2n^(2)+3n+13))/(6)`
`G(n)=sum(n^(2)+n+1)=(n(n^(2)+3n+5))/(3)`
`:.lim_(ntooo)((n(2n^(2)+3n+13))/(6))/((n(n^(2)+3n+5))/(3))=1`