The value of

$1000\left[\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + ... + \frac{1}{999 \times 1000}\right]$

• 1000

• 999

• 1001

• $\frac{1}{999}$

Six positive numbers are in GP, such that their product is 1000. If the fourth term is 1, then the last term is

• 1000

• 100

• $\frac{1}{100}$

• $\frac{1}{1000}$

Five numbers are in AP with common difference $\ne$ 0. If the 1st, 3rd and 4th terms are in GP, then

• the 5th term is always 0.

• the 1st term is always 0.

• the middle term is always 0

• the middle term is always - 2.

The sum of series

$\frac{1}{1 \times 2}{}^{25}\mathrm{C}_0 + \frac{1}{2 \times 3}{}^{25}\mathrm{C}_1 + \frac{1}{3 \times 4}{}^{25}\mathrm{C}_2 + ... + \frac{1}{26 \times 27}{}^{25}\mathrm{C}_{25}$

is

• $\frac{2^{27} - 1}{26 \times 27}$

• $\frac{2^{27} - 28}{26 \times 27}$

• $\frac{1}{2}\left(\frac{2^{26} + 1}{26 \times 27}\right)$

• $\left(\frac{2^{26} - 1}{52}\right)$

65.

Let f : R $\to$ R  be such that f is injective and f(x) f(y) = f(x + y) for all x, y $\in$if f(x), f(y) and f(z) are in GP, then x, y and z are in

• AP always

• GP always

• AP depending on the values of x, y and z

• GP depending on the values of x, y and z

If $\mathrm{P} = 1 + \frac{1}{2 \times 2} + \frac{1}{3 \times 2^2} + ... \mathrm{and} \mathrm{Q} = \frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + ...$,

then

• P = Q

• 2P =Q

• P = 2Q

• P = 4Q

If $\mathrm{x} = 1 + \frac{1}{2 \times 1!} + \frac{1}{4 \times 2!} + \frac{1}{8 \times 3!} + ...$ and y = $1 + \frac{{\mathrm{x}}^2}{1!} + \frac{{\mathrm{x}}^4}{2!} + \frac{{\mathrm{x}}^6}{3!} +\hspace{0.17em}...$ Then, the value of ${\log }_{\mathrm{e}}\left(\mathrm{y}\right)$ is

• e

• e²

• 1

• $\frac{1}{\mathrm{e}}$

The value of the infinite series

$\frac{1^2 + 2^2}{3!} + \frac{1^2 + 2^2 + 3^2}{4!} + \frac{1^2 + 2^2 + 3^2 +\hspace{0.17em}{4}^2}{5!} + ...$ is

• e

• 5e

• $\frac{5\mathrm{e}}{6} - \frac{1}{2}$

• $\frac{5\mathrm{e}}{6}$

The sum of the series $1 + \frac{1}{2}{}^{\mathrm{n}}\mathrm{C}_1 + \frac{1}{3}{}^{\mathrm{n}}\mathrm{C}_2 + ... + \frac{1}{\mathrm{n} + 1}{}^{\mathrm{n}}\mathrm{C}_{\mathrm{n}}$ is equal to

• $\frac{2^{\mathrm{n} + 1} - 1}{\mathrm{n} + 1}$

• $\frac{3\left(2^{\mathrm{n}} - 1\right)}{2\mathrm{n}}$

• $\frac{2^{\mathrm{n}} + 1}{\mathrm{n} + 1}$

• $\frac{2^{\mathrm{n}} + 1}{2\mathrm{n}}$

The value of $\sum _{\mathrm{r} = 2}^{\infty }\frac{1 + 2 + ... + \left(\mathrm{r} - 1\right)}{\mathrm{r}!}$

• e

• 2e

• $\frac{\mathrm{e}}{2}$

• $\frac{3\mathrm{e}}{2}$