Sequences and Series

If $a, b, c$ are in Arithmetic Progression, which of the following is true?

Find $a_{50}$ for the sequence $a_n = (-1)^n \cdot 2$.

The formula for the sum of the first $n$ natural numbers is:

In a Geometric Progression, $a = 4$ and $r = 0.5$. Find $a_3$.

If $a, b$ are roots of $x^2 - 3x + p = 0$ and $c, d$ are roots of $x^2 - 12x + q = 0$, where $a, b, c, d$ form a GP, then the value of $q:p$ is:

If the $3^{rd}$ term of a GP is $24$ and the $6^{th}$ term is $192$, then the $10^{th}$ term is:

The sum of all two-digit numbers which when divided by $4$ leave a remainder $1$ is:

If $a_1, a_2, \dots, a_n$ are in A.P. with common difference $d \neq 0$, then the sum $\sum_{i=1}^{n-1} \frac{1}{a_i a_{i+1}}$ is:

The sum to infinity of the series $\frac{1}{1} + \frac{1}{1+2} + \frac{1}{1+2+3} + \dots$ is:

If $(b-c)^2, (c-a)^2, (a-b)^2$ are in A.P., then $\frac{1}{b-c}, \frac{1}{c-a}, \frac{1}{a-b}$ are in:

For a finite A.P. with first term $a$ and last term $l$, the arithmetic mean of all its terms is:

If $S_n$ is the sum of first $n$ terms of an A.P., then the $n^{th}$ term $a_n$ is given by:

The $10^{th}$ term of the A.P. $-1.2, -3.2, -5.2, ...$ is:

If $x, x+3, x+6$ are the first three terms of an A.P., what is the common difference?

If the ratio of the $4^{th}$ term to the $7^{th}$ term of an A.P. is $2:3$, find the ratio of the $6^{th}$ term to the $11^{th}$ term.

If $\sum_{k=1}^n (2k+5) = 160$, then the value of $n$ is:

If the sum of first $n$ terms of an A.P. is $S_n = n^2a + n$, then the common difference is:

Let $S_1, S_2, S_3, \dots, S_p$ be the sums of the first $n$ terms of $p$ different arithmetic progressions whose first terms are $1, 2, 3, \dots, p$ and whose common differences are $1, 3, 5, \dots, (2p-1)$ respectively. The value of the sum $\sum_{k=1}^p S_k$ is equal to:

If $a_1, a_2, a_3, \dots, a_n$ are in A.P., then the value of $\sum_{i=1}^{n-1} \frac{1}{a_i a_{i+1}}$ is equivalent to:

If the sum of the first $n$ terms of an A.P. is $3n^2 + 2n$, then its $r^{th}$ term is:

Find the sum to infinity of $1 + \frac{1}{2} + \frac{1}{4} + \dots$.

Find the $4^{th}$ term of the G.P. $x, 2x, 4x, \dots$.

Determine the geometric mean of $9$ and $9$.

Which term of the G.P. $2, 8, 32, \dots$ is $512$?

The sum of an infinite G.P. is 162 and its common ratio is $8/9$. Find the first term.

If $x, y, z$ are in G.P. and $x+3, y+3, z+3$ are in A.P., then the common ratio of the G.P. is:

If the ratio of the fourth term to the second term of a G.P. is 9, what is the common ratio?

If the $p^{th}$, $q^{th}$, and $r^{th}$ terms of a non-constant arithmetic progression (A.P.) are in geometric progression (G.P.), then the common ratio of the G.P. is:

If $a, b, c$ are in G.P. and $a-b, c-a, b-c$ are in A.P., then the value of $a+4b+c$ is:

The value of $0.4232323\dots$ as a fraction is:

If $a$ and $b$ are two positive numbers, the relationship $A = G$ holds if and only if:

If the Arithmetic Mean ($A$) of two numbers is $15$, then their sum $a + b$ is:

If the Geometric Mean ($G$) of two positive numbers is $6$, then their product $ab$ is:

The Arithmetic Mean of $0.25$ and $4$ is:

If the sum of two numbers is $20$ and their Arithmetic Mean is $1.25$ times their Geometric Mean, find the two numbers.

If $a, b, c$ are positive real numbers and $abc=1$, what is the minimum value of $(1+a)(1+b)(1+c)$?

Find the minimum value of $\frac{x^2+2}{\sqrt{x^2+1}}$ for all real $x$.

If $a, b, c, d$ are positive and $a+b+c+d=1$, the maximum value of $abcd$ is:

What is the maximum value of $\sin^2 \theta \cos \theta$ for $\theta \in [0, \pi/2]$?

If $a, b, c$ are positive and $a+b+c=k$, then the minimum value of $a^2 + b^2 + c^2$ is:

What is the value of $\sum_{k=1}^{5} 4$?

If $a_n = 10n$, what is the sum of the first 2 terms?

Calculate the sum $\sum_{k=1}^{2} (k^3 - k)$.

If $a_n = n^3 - 2$, find the value of $S_2$.

Evaluate $\sum_{k=1}^n (k^2 + k + 1)$.

Find the sum to $n$ terms of $3 \cdot 8 + 6 \cdot 11 + 9 \cdot 14 + \dots$.

What is the sum of the series $1 \cdot 2^2 + 2 \cdot 3^2 + 3 \cdot 4^2 + \dots$ to $n$ terms?

Evaluate the sum to $n$ terms: $1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \dots + \frac{1}{1+2+\dots+n}$.

Sum the series whose $n$-th term is $n(n+1)(2n+1)$.

Find the sum of the series $1^2 \cdot 2 + 2^2 \cdot 3 + 3^2 \cdot 4 + \dots + n^2 \cdot (n+1)$.

If $a = 2$ and $r = 3$, find the 2nd term of the G.P.

In a G.P., $a = 1$ and $r = 2$. Find the 6th term.

Find the 3rd term of the G.P. $3, 9, 27, \dots$.

Determine the common ratio of the sequence $4, 2, 1, \dots$.

Find the 5th term of the G.P. whose $n$th term is given by $a_n = 2 \cdot 3^{n-1}$.

If $a, b, c$ are in A.P. and $a, b, d$ are in G.P., then $a, a-b, d-c$ are in:

Find the $n$th term of the G.P. $5/2, 5/4, 5/8, \dots$.

If the $n^{th}$ term of a G.P. is denoted by $T_n$, find the value of $T_{n+1} T_{n-1} / T_n^2$.

The first term of a G.P. is $1$. The sum of the third and fifth terms is $90$. Find the common ratio $r$ (assume $r > 0$).

If $S_n$ denotes the sum of the first $n$ terms of a G.P., find the value of $S_n(S_{3n} - S_{2n}) / (S_{2n} - S_n)^2$.

If the G.M. of two numbers $a$ and $b$ is $g$, what is the G.M. of $1/a$ and $1/b$?

Find the G.M. of $x$ and $x^9$.

The G.M. of $2$ and $200$ is:

If $1, G, 81$ are in G.P., what is the value of $G$?

If $x, y, z$ are positive and $y$ is the G.M. of $x$ and $z$, then find the value of $\frac{x}{y} + \frac{y}{z}$ in terms of $x$ and $y$.

Two numbers $a$ and $b$ are in the ratio $1:4$. If their G.M. is 10, find $a$.

Given $G.M. = \sqrt{27 \times 3}$. What is the value of this geometric mean?

If $G_1, G_2, \dots, G_n$ are $n$ geometric means between $a$ and $b$, then $G_1 \cdot G_n$ equals:

If $n$ G.M.s are inserted between $a$ and $b$, and $G_k$ is the $k$-th mean, find $\log(\frac{G_k}{a}) / \log(\frac{b}{a})$.

If $G$ is the G.M. between $a$ and $b$, and $A$ is the A.M., which of the following is true for $a, b > 0$ and $a \neq b$?