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

• $\frac{\mathrm{e}}{2}$

• $\frac{3\mathrm{e}}{2}$

213.

The remainder obtained when 1! + 2! + ... + 95! is divided by 15 is

• 14

• 3

• 1

• 0

If a, b and c are in arithmetic progression, then the roots of the equation ax - 2bx + c = 0 are

• $1 \mathrm{and} \frac{\mathrm{c}}{\mathrm{a}}$

• $- \frac{1}{\mathrm{a}} \mathrm{and} - \mathrm{c}$

• $- 1 \mathrm{and} - \frac{\mathrm{c}}{\mathrm{a}}$

• $- 2 \mathrm{and} - \frac{\mathrm{c}}{2\mathrm{a}}$

Let the coefficients of powers of x in the 2nd, 3rd and 4th terms in the expansion of (1 + x)ⁿ, where n is a positive integer, be in arithmetic progression. Then, the sum of the coefficients of odd powers of x in the expansion is

• 32

• 64

• 128

• 256

The sum 1 x 1! + 2 x 2! + ... + 50 x 50! equals

• 51!

• 51! + 1

• 51! + 1

• 2 $\times$ 51!

Six numbers are in AP such that their sum is 3. The first term is 4 times the third term. Then, the fifth term is

• - 15

• - 3

• 9

• - 4

The sum of the infinite series $1 +\hspace{0.17em}\frac{1}{3} + \frac{1 . 3}{3 . 6} +\hspace{0.17em}\frac{1 . 3 . 5}{3 . 6 . 9} + \frac{1 . 3 . 5 . 7}{3 . 6 . 9 . 12} + ...$ is equal to

• $\sqrt{2}$

• $\sqrt{3}$

• $\sqrt{\frac{3}{2}}$

• $\sqrt{\frac{1}{3}}$

If 64, 27, 36 are the Pth Qth and Rth terms of a GP, then P + 2Q is equal to

• R

• 2R

• 3R

• 4R

The coefficient of x¹⁰ in the expansion of 1 + (1 + x) + ... + (1 + x)²⁰

• ${}_{19}\mathrm{C}_9$

• ${}_{20}\mathrm{C}_{10}$

• ${}_{21}\mathrm{C}_{11}$

• ${}_{22}\mathrm{C}_{12}$