Question

Let $\left(a_{n}\right), n=1,2,3 \ldots$ be the sequence of real numbers such that $a_{1}=2$ and $a_{n}=\left(\frac{n+1}{n-1}\right)\left(a_{1}+a_{2}+\ldots .+a_{n-1}\right),(n \geq 2)$, then which of the following is not correct?

• A. $\left[\frac{a_{2016}}{2^{2016}}\right]=1008$
• B. $\left[\frac{a_{2016}}{2^{2016}}\right]=2016$
• C. $a_{2015}$ is divisible by $2^{2019}$
• D. If $a_{1}+a_{2}+\ldots+a_{n}>\,1008$, then least value of $n$ is 8 .
  [Where [.] denotes greatest integer function]

Answer 1

Answer: (B)

Given $a_{1}=2$ and by using given relation,
We have $a_{2}=2^{1} \cdot 3, a_{3}=2^{2} .4 a_{4}=2^{3} .5$
$\therefore a_{2016}=2^{2015} \cdot 2017$
we can also find the general term
$a_{n}=2^{n-1}(n+1)$ by using $a_{n}=s_{n}-s_{n-1}$ with no loss of generality.