Question

If $3k,$ $k,$ $[k^2-34]$ are the first three terms of a geometric progression, where $k\in R^{+}$ and $[.]$ is the greatest integer function, then the value of $\sum _{r=1}^{10}(r{)}^{k/2}$ is

• A. $55$
• B. $385$
• C. $3025$
• D. $325$

Answer 1

The correct option is C $3025$

$3k,k,[k^2-34]$ are in G.P.
$\Rightarrow k^2=3k[k^2-34]$
$\Rightarrow k=3[k^2-34]$
$\Rightarrow k$ is an integer.
$\Rightarrow k^2-34$ is also an integer.
$\Rightarrow 3k^2-k-102=0$
$\Rightarrow k=6$ or $k=-\frac{17}{3}$
As $k$ is an integer, $k=6$

Now, $\sum _{r=1}^{10}(r{)}^{k/2}=\sum _{r=1}^{10}(r{)}^3$
$=1^3+2^3+\cdots +{10}^3$
$={\left(\frac{10\times 11}{2}\right)}^2=3025$