The value of $\frac{2}{3!} +\hspace{0.17em}\frac{4}{5!} + \frac{6}{7!} + ...$

• ${\mathrm{e}}^{\frac{1}{2}}$

• e^{- 1}

• e

• ${\mathrm{e}}^{- \frac{1}{3}}$

If sum of an infinite geometric series is 4/3 and its Ist term is 3/4, then its common ratio is

• 7/16

• 9/16

• 1/9

• 7/9

The value of $\frac{{\log }_3\left(5\right) \times {\log }_{25}\left(27\right) \times {\log }_{49}\left(7\right)}{{\log }_{81}\left(3\right)}$ is

• 1

• 6

• $\frac{2}{3}$

• 3

Sum of n terms of the following series 1³ + 3³ + 5³ + 7³ + ... is

• n²(2n² - 1)

• n³(n - 1)

• n³ + 8n + 4

• 2n⁴ + 3n²

GM and HM of two numbers are 10 and 8 respectively. The numbers are

• 5, 20

• 4, 25

• 2, 50

• 1,100

The value of n for which is the $\frac{{\mathrm{x}}^{\mathrm{n} +\hspace{0.17em}1} + {\mathrm{y}}^{\mathrm{n} + 1}}{{\mathrm{x}}^{\mathrm{n}} + {\mathrm{y}}^{\mathrm{n}}}$ geometric mean of x and y is

• n = - $\frac{1}{2}$

• n = $\frac{1}{2}$

• n = 1

• n = - 1

If angles A, B and C are in AP, then $\frac{\mathrm{a} + \mathrm{c}}{\mathrm{b}}$ equal to

• 2$\sin \left(\frac{\mathrm{A} - \mathrm{C}}{2}\right)$

• $2\cos \left(\frac{\mathrm{A} - \mathrm{C}}{2}\right)$

• $\cos \left(\frac{\mathrm{A} - \mathrm{C}}{2}\right)$

• $\sin \left(\frac{\mathrm{A} - \mathrm{C}}{2}\right)$

The sum of the infinite series

$1 + \frac{1}{2!} + \frac{1 . 3}{4!} + \frac{1 . 3. 5}{6!} + ...$

• e

• e²

• $\sqrt{\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

If three positive real numbers a, b, c are in AP and abc = 4, then the minimum possible value of b is 2

• $2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

• $2^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

• $2^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

For what value of m, $\frac{{\mathrm{a}}^{\mathrm{m} + 1} + {\mathrm{b}}^{\mathrm{m} + 1}}{{\mathrm{a}}^{\mathrm{m}} + {\mathrm{b}}^{\mathrm{m}}}$  is the arithmetic mean of 'a' and 'b' ?

• 1

• 0

• 2

• None of these