Find the limit of the sequence.
$\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

A) $\frac { 4 } { 5 }$
B) $\frac { 5 } { 4 }$
C)3
D)4
E)does not exist

Question

Find the limit of the sequence. 
 $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$

A) $\frac { 4 } { 5 }$
B) $\frac { 5 } { 4 }$
C)3
D)4
E)does not exist

Quizplus · Accepted Answer

The given sequence is $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$. Simplifying inside the brackets first, we get $\frac { n ( n + 3 ) } { 4 }$. When this is simplified further, it becomes $\frac { n^2 + 3n } { 4 }$. The entire expression then simplifies to $1 + \frac { 3 } { 4 }$ as $n$ approaches infinity, since the $1/n$ term outside the brackets and the $n$ in the denominator inside the brackets cancel out. Thus, the limit of the sequence as $n$ approaches infinity is $\frac { 5 } { 4 }$.

Find the Limit of the Sequence 1n(n+1n[n(n+3)4])\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)n1​(n+n1​[4n(n+3)​])

Find the Limit of the Sequence $\frac { 1 } { n } \left( n + \frac { 1 } { n } \left[ \frac { n ( n + 3 ) } { 4 } \right] \right)$