The integer just greater than (3+√5)(2 n) is divisible by, (n ∈ N),

Question

The integer just greater than (3+√5)(2 n) is divisible by, (n ∈ N), (A) 2n -1 (B) 2n + 1 (C) 2n + 2 (D) Not divisible by 2

Tardigrade Admin · Accepted Answer

R=(3+√5)2 n, G=(3-√5)2 n Let [R]+1=I ( [ ] greatest integer function). ⇒ R + G = I (∵ 0< G < l ) ⇒ (3+√5)2 n+(3-√5)2 n=1 Seeing the option put n=1 ⇒ I =28 is divisible by 4 i.e 2 n +1

Q. The integer just greater than $(3 + 5)^{(2 n)}$ is divisible by, $(n \in N)$ ,

$2^{n - 1}$

$2^{n + 1}$

$2^{n + 2}$

Not divisible by 2

$R = (3 + 5)^{2 n}, G = (3 - 5)^{2 n}$
Let $[R] + 1 = I$ ( [ ] greatest integer function).
$\Rightarrow R + G = I (∵ 0 < G < l)$
$\Rightarrow (3 + 5)^{2 n} + (3 - 5)^{2 n} = 1$
Seeing the option put $n = 1$
$\Rightarrow I = 28$ is divisible by $4$ i.e $2^{n + 1}$

Q. The integer just greater than (3+5​)(2n) is divisible by, (n∈N),

2n−1

2n+1

2n+2