31.

The sum of the first n terms of the series  when n is even. When n is odd the sum is

• 
• 
• 
• 

The sum of series 

Suppose four distinct positive numbers a₁, a₂, a3, a₄ are in G.P. Let b₁ = a₁, b₂ = b₁ + a₂, b₃ = b₂ + a₃ and b₄ = b₃ + a₄.
Statement-I: The numbers b₁, b₂, b₃, b₄are neither in A.P. nor in G.P. Statement-II: The numbers b₁, b₂, b₃, b₄ are in H.P.

• Both statement-I and statement-II are true but statement-II is not the correct explanation of statement-I

• Both statement-I and statement-II are true, and statement-II is correct explanation of Statement-I

• Statement-I is true but statement-II is false.

• Statement-I is true but statement-II is false.

Let a₁, a₂, a₃, ...., a₄₉ be in A.P. such that $$\begin{gathered} \sum _{k = 0}^{12}{a}_{k-1} = 416 and a_9 + a_{43} = 66. \\ if a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m then m is equal to \end{gathered}$$

• 33

• 66

• 68

• 34

Let f : R ➔ R be such that f is injective and f(x)f(y) = f(x + y) for $⟇$ x, y $\in$ R. If f(x), f(y), f(z) are in G.P., then x, y, z are in

• AP always

• GP always

• AP depending on the value of x, y, z

• GP depending on the value of x, y, z

In a GP series consisting of positive terms, each term is equal to the sum of next two terms. Then, the common ratio of this GP series is

• $\sqrt{5}$

• $\frac{\sqrt{5} - 1}{2}$

• $\frac{\sqrt{5}}{2}$

• $\frac{\sqrt{5} +\hspace{0.17em}1}{2}$

If x is a positive real number different from 1 such that ${\log }_{\mathrm{a}}\left(\mathrm{x}\right), {\log }_{\mathrm{b}}\left(\mathrm{x}\right), {\log }_{\mathrm{c}}\left(\mathrm{x}\right)$ are in AP, then

• $\mathrm{b} = \frac{\mathrm{a} + \mathrm{c}}{2}$

• $\sqrt{\mathrm{ac}}$

• ${\mathrm{c}}^2 = {\left(\mathrm{ac}\right)}^{{\log }_{\mathrm{a}}\left(\mathrm{b}\right)}$

• None of these

If a, x are real numbers and $\left|\mathrm{a}\right|$ < 1, $\left|\mathrm{x}\right|$ < 1, then 1 + (1+ a) x + (1+ a + a²)x² + ...$\infty$ is equal to

• $\frac{1}{\left(1 - \mathrm{a}\right)\left(1 - \mathrm{ax}\right)}$

• $\frac{1}{\left(1 - \mathrm{a}\right)\left(1 - \mathrm{x}\right)}$

• $\frac{1}{\left(1 - \mathrm{x}\right)\left(1 - \mathrm{ax}\right)}$

• $\frac{1}{\left(1 + \mathrm{ax}\right)\left(1 - \mathrm{a}\right)}$

If ${\log }_{0.3}\left(\mathrm{x} - 1\right) < {\log }_{0.09}\left(\mathrm{x} - 1\right)$, then x lies in the interval

• $\left(2, \infty \right)$

• (1, 2)

• (- 2, - 1)

• None of these

The value of $\sum _{\mathrm{n} = 1}^{13}\left({\mathrm{i}}^{\mathrm{n}} + {\mathrm{i}}^{\mathrm{n} + 1}\right), \mathrm{i} = \sqrt{- 1}$ is

• i

• i - 1

• 1

• 0