Let (an), n=1,2,3 ldots be the sequence of real numbers such that a1=2 and an=((n+1/n-1))(a1+a2+ ldots .+an-1),(n ≥ 2), then which of the following is not correct?

Question

Let (an), n=1,2,3 ldots be the sequence of real numbers such that a1=2 and an=((n+1/n-1))(a1+a2+ ldots .+an-1),(n ≥ 2), then which of the following is not correct? (A) [(a2016/22016)]=1008 (B) [(a2016/22016)]=2016 (C) a2015 is divisible by 22019 (D) If a1+a2+ ldots+an> 1008, then least value of n is 8 . [Where [.] denotes greatest integer function]

Tardigrade Admin · Accepted Answer

Given a1=2 and by using given relation, We have a2=21 ⋅ 3, a3=22 .4 a4=23 .5 ∴ a2016=22015 ⋅ 2017 we can also find the general term an=2n-1(n+1) by using an=sn-sn-1 with no loss of generality.

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

$[\frac{a _{2016}}{2 ^{2016}}] = 1008$

$[\frac{a _{2016}}{2 ^{2016}}] = 2016$

$a_{2015}$ is divisible by $2^{2019}$

If $a_{1} + a_{2} + \dots + a_{n} > 1008$ , then least value of $n$ is 8 .
[Where [.] denotes greatest integer function]

Given $a_{1} = 2$ and by using given relation,
We have $a_{2} = 2^{1} \cdot 3, a_{3} = 2^{2} .4 a_{4} = 2^{3} .5$
$∴ a_{2016} = 2^{2015} \cdot 2017$
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.

Q. Let (an​),n=1,2,3… be the sequence of real numbers such that a1​=2 and an​=(n−1n+1​)(a1​+a2​+….+an−1​),(n≥2), then which of the following is not correct?

[22016a2016​​]=1008

[22016a2016​​]=2016

a2015​ is divisible by 22019

If a1​+a2​+…+an​>1008, then least value of n is 8 . [Where [.] denotes greatest integer function]