Question

Assertion (A) : $\left[\frac{1}{2}\right]+[\frac{1}{2}+\frac{1}{1000}]+[\frac{1}{2}+\frac{2}{1000}]+[\frac{1}{2}+\frac{3}{1000}]+...+[\frac{1}{2}+\frac{999}{1000}]=500$

(where [.] denotes G.I.F.)

Reason (R) : $[\frac{1}{2}+\frac{r}{1000}]=\{\begin{array}{cc}0;& \text{if}0\le r<500\\ 1;& \text{if}500\le r\le 999\end{array}$

• A. Both A and R are individually true and R is the correct explanation of A
• B. Both A and R are individually true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

Answer: (A)

Both A and R are individually true and R is the correct explanation of A

$\left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{1000}\right]+[\frac{1}{2}+\frac{2}{1000}]+\dots .+[\frac{1}{2}+\frac{999}{1000}]$

$=\underset{–}{\left[\frac{1}{2}\right]+\left[\frac{1}{2}+\frac{1}{1000}\right]+[\frac{1}{2}+\frac{2}{1000}]+....+[\frac{1}{2}+\frac{499}{1000}]}+\underset{–}{[\frac{1}{2}+\frac{500}{1000}]+[\frac{1}{2}+\frac{501}{1000}].........+[\frac{1}{2}+\frac{999}{1000}]}$
$500\text{terms}$ $500\text{terms}$

First $500$ terms will vanish individually since they lie between $0$ and $1$

And next $500$ terms will be $1$ individually since they lie between $1$ and $2$

Hence required summation is $500$.

Reason is obviously correct and explaining the Assertion