sequences and series

If a, b, c are in HP, then $\frac{\mathrm{a}}{\mathrm{b} + \mathrm{c}}, \frac{\mathrm{b}}{\mathrm{c} + \mathrm{a}}, \frac{\mathrm{c}}{\mathrm{a} + \mathrm{b}}$ will be in

• AP

• GP

• HP

• None of these

The sum of the coefficients of (6a - 5b)ⁿ, where n is a positive integer, is

• 1

• - 1

• 2ⁿ

• 2^{n - 1}

If a^x = b^y = c^z = d^u and a, b, c, d are in GP, then x, y, z, u are in

• AP

• GP

• HP

• None of these

If x is numerically so small so that x² and higher power of x can be neglected, then ${\left(1 + \frac{2\mathrm{x}}{3}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} . {\left(32 + 5\mathrm{x}\right)}^{- \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}$ is approximately equal to

• $\frac{32 + 31\mathrm{x}}{64}$

• $\frac{31 + 32\mathrm{x}}{64}$

• $\frac{31 - 32\mathrm{x}}{64}$

• $\frac{1 - 2\mathrm{x}}{64}$

If the sides of a right angle triangle form an AP, the 'sin' of the acute angles are

• $\left(\frac{3}{5}, \frac{4}{5}\right)$

• $\left(\sqrt{3}, \frac{1}{\sqrt{3}}\right)$

• $\left(\sqrt{\frac{\sqrt{5} - 1}{2}}, \sqrt{\frac{\sqrt{5} - 1}{2}}\right)$

• $\left(\sqrt{\frac{\sqrt{3} - 1}{2}}, \sqrt{\frac{\sqrt{3} - 1}{2}}\right)$

If H is the harmonic mean between P and Q, then the value of $\frac{\mathrm{H}}{\mathrm{P}} + \frac{\mathrm{H}}{\mathrm{Q}}$ is

• 2

• $\frac{\mathrm{PQ}}{\mathrm{P} +\hspace{0.17em}\mathrm{Q}}$

• $\frac{1}{2}$

• $\frac{\mathrm{P} +\hspace{0.17em}\mathrm{Q}}{\mathrm{PQ}}$

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

• e

• 2e

• e²

• $\frac{1}{\mathrm{e}}$

If a₁, a₂, a₃ ..., a_n are in AP, where a_i > 0 for all i. Find the sum of series $\frac{1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_2}} + \frac{1}{\sqrt{{\mathrm{a}}_2} + \sqrt{{\mathrm{a}}_3}} + \frac{1}{\sqrt{{\mathrm{a}}_3} + \sqrt{{\mathrm{a}}_4}} + ... + \frac{1}{\sqrt{{\mathrm{a}}_{\mathrm{n} - 1}} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} + 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} - 1}{\sqrt{{\mathrm{a}}_1} - \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} + 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

• $\frac{\mathrm{n} - 1}{\sqrt{{\mathrm{a}}_1} + \sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

If S₁, S₂ and S₃ are the sums of n, 2n and 3n terms of an arithmetic progression respectively, then

• S₂ = 3S₃ - 2S₁

• S₃ = 4(S₁ + S₂)

• S₃ = 3(S₂ - S₁)

• S₃ = 2(S₂ + S₁)

$\frac{{\mathrm{C}}_0}{1} + \frac{{\mathrm{C}}_2}{3} + \frac{{\mathrm{C}}_4}{5} + \frac{{\mathrm{C}}_6}{7} + ...$ is equal to

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

• $\frac{2^{\mathrm{n} + 1}}{\mathrm{n} + 3}$

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

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