Question

Let $S={\sum }_{k=1}^{80}\frac{1}{\sqrt{K}.}$. Then [S], where [.] denotes the greatest integer function,equals

• A. $17$
• B. $16$
• C. $15$
• D. $14$

Answer 1

The correct option is A $17$

$S={\sum }_{K=1}^{80}\frac{1}{\sqrt{K}}$

$S=2{\sum }_{K=1}^{80}\frac{1}{2\sqrt{K}}$

$S<2{\sum }_{K=1}^{80}\frac{1}{\sqrt{K}+\sqrt{K-1}}$

$=2{\sum }_{K=1}^{80}\frac{\sqrt{K}-\sqrt{K-1}}{(\sqrt{K}+\sqrt{K-1})(\sqrt{K}-\sqrt{K-1})}$

$=2{\sum }_{K=1}^{80}\frac{\sqrt{K}-\sqrt{K-1}}{1}$

$=2\sqrt{80}$

$\therefore \left[S\right]=\left[17.88\right]=17$

Option A