Mathematics - Sequence & Series Question with Solution | TestHub

By strict AM-GM, $n+S_n={\sum }_{k=1}^n(1+1/k)>n{\left({\prod }_{k=1}^n(1+1/k)\right)}^{1/n}=n(n+1{)}^{1/n}$. Hence option (A) is true. Option (D) also follows: its right side is $n(n+1{)}^{1/n}/(n+5)<2n/(n+5)<2$ ; for $n\ge 4,S_n\ge S_4=25/12>2$; at $n=3$, the right side is $3\cdot 4^{1/3}/8<1<S_3$.