Mock Test 2 — Sequence and Series

If $\log_3 2, \log_3 (2^x - 5),$ and $\log_3 (2^x - \frac{7}{2})$ form a valid Arithmetic Progression, then the value of the variable $x$ is:

Let $a, b, c$ be positive real numbers. If $a, b, c$ form an AP, $b, c, a$ form a GP, then $c, a, b$ must form an:

The sum to infinity of the convergent series $\frac{3}{1^2 \cdot 2^2} + \frac{5}{2^2 \cdot 3^2} + \frac{7}{3^2 \cdot 4^2} + \dots$ evaluated using telescoping partial fractions is:

If $a, b, c$ are positive real numbers such that $a + b + c = 18$, the maximum possible value of the product expression $a^2 b^3 c^1$ found using weighted AM--GM inequalities is:

The sum of the first $n$ terms of the series $1 + (1+d)r + (1+2d)r^2 + \dots$ is an AGP. If $S_\infty = \frac{35}{4}$ when $r = \frac{1}{5}$, the value of the common difference $d$ is:

If $S_1, S_2, S_3$ are the sums of the first $n, 2n, 3n$ terms of an arithmetic progression respectively, then the value of the ratio expression $\frac{S_3}{S_2 - S_1}$ is always:

The value of the sum series $\sum_{k=1}^n \frac{1}{k(k+1)(k+2)}$ evaluated using the $V_n$ method is:

If the Arithmetic Mean between two positive numbers $a$ and $b$ is 9 and their Geometric Mean is 4, then the numbers are the roots of which quadratic equation?

The sum to infinity of the series $\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + \dots$ is:

If $a, b, c$ form a valid Harmonic Progression, then the expression $\frac{1}{b-a} + \frac{1}{b-c}$ is identically equal to:

Find the value of the linear sum $\sum_{k=1}^5 (k^3 - 3k^2)$.

If the first three terms of a convergent geometric progression are $x, x+2, x+6$, find the value of the negative integer $x$.

If $\text{AM} \times \text{HM} = 9$ for two positive numbers, find the value of their Geometric Mean (GM).

Assertion (A): The sum of the series $\sum_{k=1}^n (k^3 + 2k)$ can be evaluated by splitting it into separate standard polynomial sums.
Reason (R): Summation operators satisfy linearity properties, allowing them to distribute across addition and preserve constant scalar multipliers.

Assertion (A): If $a, b, c$ are in an AP, then $e^a, e^b, e^c$ must form a GP.
Reason (R): For an arithmetic progression, $2b = a+c$. Raising a base constant to these powers matches the geometric progression definition: $(e^b)^2 = e^{2b} = e^{a+c} = e^a \cdot e^c$.