Mock Test 1 — Sequence and Series

If the sum of the first $n$ terms of an AP is given by $S_n = 2n^2 + 3n$, then the common difference $d$ of this arithmetic sequence is:

If three positive numbers $a, b, c$ form a geometric progression, then the equations $a x^2 + 2bx + c = 0$ and $d x^2 + 2ex + f = 0$ share a common root if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ form an:

The sum to infinity of the convergent geometric progression $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ is exactly:

If we insert 5 arithmetic means between 2 and 20, the common difference $d$ of the resulting progression is:

The value of the geometric mean (GM) between the numbers 4 and 16 is:

If $a, b, c$ are in an arithmetic progression, then the terms $\frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab}$ must form an:

The sum to infinity of the convergent AGP series $1 + \frac{2}{2} + \frac{3}{4} + \frac{4}{8} + \dots$ evaluates to:

The value of the linear sum $\sum_{k=1}^{50} k$ tracking the first 50 natural numbers is:

If $x > 0$, the minimum possible value of the expression $2x + \frac{3}{x}$ found using mean inequalities is:

The $n^{\text{th}}$ term of a Harmonic Progression whose first two terms are $\frac{1}{2}$ and $\frac{1}{5}$ is given by:

If the sum of the squares of the first $n$ natural numbers $\sum_{k=1}^n k^2$ equals 55, find the value of $n$.

Find the value of the common ratio $r$ for an infinite geometric progression if its first term is 3 and its sum to infinity converges to 4.

Determine the number of geometric means that must be inserted between 1 and 64 such that the common ratio of the resulting progression is exactly 2.

Assertion (A): The sum of an infinite geometric series $1 + r + r^2 + \dots$ can only be evaluated if the common ratio satisfies $|r| < 1$.
Reason (R): If the absolute value of the common ratio is greater than or equal to 1 ($|r| \ge 1$), the terms grow continuously or oscillate, causing the infinite series to diverge instead of converging to a single limiting value.

Assertion (A): For any two positive distinct real numbers, their Arithmetic Mean is strictly greater than their Geometric Mean ($\text{AM} > \text{GM}$).
Reason (R): The algebraic identity $\frac{a+b}{2} - \sqrt{ab} = \frac{1}{2}(\sqrt{a} - \sqrt{b})^2$ is a perfect square expression, which is strictly positive for any distinct positive numbers.