Exploring some more Progressions

What is the common difference of the AP: $a-b, a, a+b, \dots$?

The general term of a sequence is $a_n = (-1)^n \times n$. What is the $4^{th}$ term?

If the first term is $5$ and the common ratio is $2$, what is the $3^{rd}$ term of the Geometric Progression?

Which of the following is NOT an AP?

A sequence is defined by the general term $a_n = \frac{1}{n^2 + n}$. What is the sum of the first $20$ terms of this sequence?

What is the $n^{th}$ term of the sequence: $0, 3, 8, 15, \dots$?

Find the value of $x$ if $x+1, 3x, 4x+2$ are in AP.

If $A$ is the arithmetic mean and $G$ is the geometric mean between two positive numbers, then the numbers are:

The number of terms of the AP $24, 21, 18, ...$ that must be taken so that their sum is 78 is:

Find the sum to $n$ terms of the series $1 \times 2 + 2 \times 3 + 3 \times 4 + ...$

What is the sum of the first $3$ terms of the GP: $2, 4, 8, \dots$?

Calculate the sum of the first $4$ terms of the GP: $100, 10, 1, 0.1$.

Find the sum of the first $3$ terms of the GP where $a = 1$ and $r = -1$.

If $a = 1$ and $r = 4$, find $S_5$.

Find the sum of the first $n$ terms of a GP if $a=5$ and $r=1$.

The sum of the first $n$ terms of the progression $6, 12, 24, \dots$ is 1530. Find $n$.

If $S_n$ denotes the sum of the first $n$ terms of a GP, find $\frac{S_{2n}}{S_n}$ in terms of the common ratio $r$.

If the sum of $n$ terms of a GP is $S_n = \frac{x^n - 1}{x - 1}$ and $x = 2$, find $n$ such that $S_n = 2047$.

The sum of first $n$ terms of the series $1 - a + a^2 - a^3 + \dots$ ($a \neq -1$) is:

A geometric progression has first term $a$ and common ratio $r$. If the sum of the first $2n$ terms is $25$ times the sum of the first $n$ terms, and $r^n > 0$, find $r^n$.

The sum of an infinite GP is 50. If the first term is 10, find the common ratio.

What is the sum of $7 + 0.7 + 0.07 + \dots$?

Find the sum of the infinite GP: $25, 5, 1, \dots$

Determine the sum of the series $1 - \frac{1}{4} + \frac{1}{16} - \dots$

If the first term of an infinite GP is $a=2$, what range of values can the sum $S$ take if $0 < r < 1$?

In an infinite GP, the first term is 1 and each term is the sum of all subsequent terms. What is the common ratio?

An equilateral triangle is drawn with side $a$. A second equilateral triangle is formed by joining the midpoints of the sides of the first. This is repeated infinitely. Find the sum of the perimeters of all triangles.

Find the sum of the series $1 + 2(1/3) + 3(1/3)^2 + 4(1/3)^3 + \dots$ to infinity.

A sequence is defined by $a_{n+1} = \frac{1}{3} a_n$ with $a_1 = 10$. Find the sum of all terms in the sequence.

If $x = 0.999\dots$, prove using infinite GP that $x = 1$. What is the common ratio used in this proof?

What is the sum of $1! + 2! + 3!$?

In the sequence $T_n = n(n+1)$, what is the $2^{nd}$ term?

If you have 4 different colored flags, how many different signals can you make using 1 flag?

What is the next term in the sequence $1, 3, 7, 13, 21, \dots$?

If the number of permutations of $n$ distinct objects taken $r$ at a time is $^nP_r = 1680$ and the number of combinations of the same $n$ objects taken $r$ at a time is $^nC_r = 70$, find the value of $n$.

In how many ways can $2$ students be chosen from a group of $5$ to be the President and Vice-President?

The $n^{th}$ term of the sequence $0, 3, 8, 15, \dots$ is given by:

How many paths are there from $(0,0)$ to $(4,4)$ on a grid if you can only move right or up?

Find the value of $n$ if $\binom{n}{2} = 45$.

In how many ways can 5 distinct rings be worn on 4 fingers of the right hand?

Find $t_n$ for the sequence $1, 8, 15, 22, \dots$.

If $t_n = 9n - 5$, find $t_2$.

In the sequence $20, 40, 60, 80, \dots$, what is $t_n$?

Find $t_n$ for the progression $3, 3, 3, 3, \dots$.

What is $t_n$ for the sequence $1, 8, 21, 40, \dots$?

Find the first term ($t_1$) for the sequence where $t_n = 2n^2 - 5n + 4$.

Determine the general term for the sequence: $100, 97, 94, 91, \dots$

Find the $10^{th}$ term of the sequence $1 \times 2, 2 \times 3, 3 \times 4, \dots$.

Determine the general term of $0, 5, 18, 39, 68, \dots$ using the method of differences.

Calculate the sum of the first $20$ terms of the sequence where $t_n = 2n - 1$.

Find the sum of the first 4 terms of the sequence $2, 4, 8, 16$.

If $a_n = 5n + 2$, find the sum of the first 2 terms.

What is the sum of the first 10 multiples of 1?

Calculate the sum: $\sum_{k=1}^{3} k^2$.

Find the sum of the first 100 natural numbers and then divide by 5.

If $S_n$ is the sum of the first $n$ odd numbers and $K_n$ is the sum of the first $n$ even numbers, find $K_{10} - S_{10}$.

Calculate the sum of $5^2 + 6^2 + 7^2 + 8^2$.

Calculate the sum of the series $\sum_{r=1}^n \frac{1}{(3r-2)(3r+1)}$.

Find the sum of $n$ terms of the series whose $n$-th term is $n^2 + 2^n$.

Find the sum of the series $1 \cdot 5 + 2 \cdot 6 + 3 \cdot 7 + \dots$ to $n$ terms.