sequences and series

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

If i = $\sqrt{- 1}$ and n is a positive integer, then iⁿ + i^{n + 1} + i^{n + 3} is equal to

• 1

• i

• iⁿ

• 0

If a, b, c are in GP (a> 1, b > 1, c > 1), then for any real number x (with x > 0, x $\ne$ 1), ${\log }_{\mathrm{a}}\left(\mathrm{x}\right), {\log }_{\mathrm{b}}\left(\mathrm{x}\right), {\log }_{\mathrm{c}}\left(\mathrm{x}\right)$ are in

• GP

• AP

• HP

• GP but not in HP

If $\mathrm{x} = {\log }_{\mathrm{a}}\left(\mathrm{bc}\right), \mathrm{y} = {\log }_{\mathrm{b}}\left(\mathrm{ca}\right), \mathrm{z} = {\log }_{\mathrm{c}}\left(\mathrm{ab}\right)$,  then the value of $\frac{1}{1 + \mathrm{x}} + \frac{1}{1 + \mathrm{y}} + \frac{1}{1 + \mathrm{z}}$ will be

• x + y + z

• 1

• ab + bc + ca

• abc

Find the sum of the first n terms of the series 0.2 + 0.22 + 0.222 + ...