Question

If [.] denotes the greatest integer function, then the value of natural number $n$ satisfying the equation
$\left[{\log }_{2}1\right]+\left[{\log }_{2}2\right]+\left[{\log }_{2}3\right]+\cdots +\left[{\log }_{2}n\right]=1538$ is

Answer 1

If $2^{r-1}\le k<2^r,$ where $r$ is an integer, then
$r-1\le {\log }_{2}k<r\Rightarrow \left[{\log }_{2}k\right]=r-1$
We have,
$\left[{\log }_{2}1\right]+\left[{\log }_{2}2\right]+\left[{\log }_{2}3\right]+\cdots +\left[{\log }_{2}n\right]=1538$

$\Rightarrow \left[{\log }_2{2}^0\right]+\sum _{k=2}^{2^2-1}\left[{\log }_{2}k\right]+\sum _{k=2^2}^{2^3-1}\left[{\log }_{2}k\right]+\sum _{k=2^3}^{2^4-1}\left[{\log }_{2}k\right]+\cdots =1538$

$\Rightarrow 0+\sum _{k=2}^{2^2-1}1+\sum _{k=2^2}^{2^3-1}2+\sum _{k=2^3}^{2^4-1}3+\sum _{k=2^4}^{2^5-1}4+\cdots =1538$

$\Rightarrow 0+1\times 2+2\times 2^2+3\times 2^3+4\times 2^4+\cdots =1538$

We observe that,
$0+1\times 2+2\times 2^2+3\times 2^3+4\times 2^4+5\times 2^5+6\times 2^6+4\times 2^7$
$=0+2+8+24+64+160+384+896=1538$

$\Rightarrow n=1+2+2^2+2^3+2^4+2^5+2^6+2^7\therefore n=\left(\frac{2^8-1}{2-1}\right)=255$