KVPY Sample Paper KVPY Stream-SX Model Paper-28

For \[n=\text{ }1,2,3....\]the value of \[\left[ \frac{n+1}{2} \right]+\left[ \frac{n+2}{4} \right]+\left[ \frac{n+4}{8} \right]+\left[ \frac{n+8}{16} \right]+......=\] (Where [.] denotes the greatest integer function)

A) \[n-1\]

B) n

C) \[n+2\]

D) 2n

Correct Answer: B

Solution :

We have \[n=\left[ \frac{n}{2} \right]+\left[ \frac{n+1}{2} \right]\]or

\[n=\left[ \frac{n+1}{2} \right]+\left[ \frac{n}{4} \right]+\left[ \frac{n+2}{4} \right]\]

(Using (1) again and again)

\[=\left[ \frac{n+1}{2} \right]+\left[ \frac{n+2}{4} \right]+\left[ \frac{n+4}{8} \right]+.....\]