Sequences and Progressions

In the sequence $1, 3, 9, 27, \dots$, what is the common ratio?

What is the next term in the sequence: $4, 9, 14, 19, \dots$?

Predict the next term: $20, 17, 14, 11, \dots$?

Identify the next term: $1/5, 2/5, 3/5, 4/5, \dots$?

Consider the sequence $1, 2, 4, 8, 16, \dots$. What is the $7^{th}$ term?

Find the next term in the sequence $10, 20, 35, 55, 80, \dots$ where the difference increases by $5$ each time ($10, 15, 20, 25$).

In the sequence $0, 3, 8, 15, 24, \dots$, the rule is $n^2 - 1$. What is the next term?

Determine the $6^{th}$ term of the sequence: $1/2, 2/5, 3/10, 4/17, \dots$

Identify the pattern and find the next term: $1, 5, 13, 25, 41, \dots$

Find the $5^{th}$ term of the sequence where $a_1=10$ and $a_n = a_{n-1} - 2n$ for $n > 1$.

If $a_1 = 2$ and $a_n = a_{n-1} + 3$, what is $a_4$?

Find the explicit rule for $1, 8, 27, 64, \dots$.

What is the next term in the sequence $2, 6, 10, 14, \dots$?

If $a_n = 5n - 5$, what is $a_1$?

Find the rule for the sequence $0, 2, 6, 12, 20, \dots$

Identify the 4th term of the sequence $a_n = \frac{2^n}{n}$.

If $a_n = n!$ (where $n! = n \cdot (n-1) \dots 1$), find $a_5$.

For the sequence $a_n = \frac{(-1)^n}{n}$, find the value of $a_3 - a_2$.

If a sequence is defined by $a_n = \frac{1}{2}n(n+1)(n+2)$, calculate $a_3$.

Find $a_4$ for the sequence $a_n = \sum_{k=1}^n k^2$.

Find the $20^{th}$ term of the AP whose $a = -3$ and $d = 4$.

The first term of an AP is $p$ and the common difference is $q$. What is the $10^{th}$ term?

If $a_2 = 13$ and $a_5 = 25$, find the common difference $d$.

What is the $8^{th}$ term of the AP: $x, x+2, x+4, \dots$?

A manufacturer of electric bicycles produced $150$ units in the $4^{th}$ year and $330$ units in the $10^{th}$ year. Assuming that the production increases uniformly by a fixed number every year, calculate the number of units produced in the $15^{th}$ year.

Find the $n^{th}$ term of the AP: $5, 2, -1, -4, -7, \dots$.

How many multiples of $7$ are there between $100$ and $300$?

If $S_n$ denotes the sum of first $n$ terms of an AP, find $a_{10}$ if $S_n = 2n^2 + 5n$.

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. If the top and bottom rungs are $2\frac{1}{2}$ m apart, find the length of the 6th rung from the bottom.

Find the middle term of the AP $7, 13, 19, \dots, 241$.

What is the sum of the first 39 natural numbers?

Find the sum of the first 38 natural numbers.

Calculate the sum of the first 37 natural numbers.

Find the value of $1+2+3+...+36$.

Let $S_n$ represent the sum of the first $n$ natural numbers. If the relationship $S_{2n} = 3S_n + 6$ holds true for a certain natural number $n$, what is the value of $n$?

A theater is designed such that the first row has $1$ seat, the second row has $2$ seats, the third row has $3$ seats, and so on. If the total seating capacity of the theater is $435$ seats, find the total number of rows in the theater.

Determine $n$ if the sum of the first $n$ natural numbers is 253.

Find the value of $n$ such that $1 + 2 + 3 + ... + n = n^2 - 55$.

If the sum of the first $n$ natural numbers is $x$, and the sum of the first $2n$ natural numbers is $x + 100$, find $n$.

If the sum of the first $n$ natural numbers is $6n$, what is the value of $n$?

If a bank account balance doubles every year, what is the common ratio $r$ of the geometric progression representing the yearly balances?

Find the $4^{th}$ term of the geometric sequence: $1, 4, 16, \dots$

In a GP, if $a = \frac{1}{10}$ and $r = 10$, find the $2^{nd}$ term.

What is the common ratio of the sequence: $18, 6, 2, \frac{2}{3}, \dots$?

Find the $7^{th}$ term of the GP $4, -12, 36, -108, \dots$.

A car's value decreases to half every $3$ years. If it is worth $₹8,00,000$ now, what will it be worth after $9$ years?

If the first term of a GP is $10$ and the common ratio is $0.1$, find the $4^{th}$ term.

If $a, b, c$ are in GP, then $\frac{a^2+ab+b^2}{bc+c^2+ca}$ is equal to:

A sequence is defined by $a_1 = 3$ and $a_{n+1} = 2a_n$. Find the $10^{th}$ term.

If the $p^{th}$ term of a GP is $A$ and the $q^{th}$ term is $B$, find the common ratio $r$.

The number of moves for Tower of Hanoi for 1, 2, 3, 4 disks are 1, 3, 7, 15. This sequence is called a:

In a sequence where $a_n = 3^n$, what is $a_4 / a_2$?

In the Sierpinski Triangle, the number of non-shaded (empty) triangles added at Stage 1 is:

If you need 255 moves to solve a Tower of Hanoi puzzle, how many disks are you using?

What is the common ratio of the geometric progression formed by the number of segments removed at each stage of the Sierpinski Triangle?

In the Tower of Hanoi, if you double the number of disks from $3$ to $6$, the number of moves increases from $7$ to:

The perimeter of the Koch Snowflake at Stage $0$ is $P$. What is the perimeter at Stage $1$?

In the Tower of Hanoi, if $n=4$, how many total moves involve Disk 3?

What is the sum of the infinite geometric series representing the total length of all segments in the Cantor set? ($1 + 2/3 + 4/9 + ...$ is NOT the length, rather the remaining segments are $L_n = (2/3)^n$. Sum all $L_n$ from $n=0$ to $\infty$).

For a Tower of Hanoi with $n=10$, which disk is moved at step $512$?