Predicting What Comes Next: Exploring Sequences

What is the next term in the halving sequence $64, 32, 16, 8, \dots$?

Identify the next term in the constant sequence $2, 2, 2, 2, \dots$.

What is the next term in the sequence $8, 16, 24, 32, \dots$?

Identify the next term in the sequence $1.1, 2.2, 3.3, 4.4, \dots$.

Find the next term in the sequence $1, 5, 9, 13, \dots$.

Identify the $10^{th}$ term of the sequence where $a_n = \frac{n+5}{2}$.

If the $n^{th}$ term is $a_n = 2n^2 - 1$, find $a_3$.

Determine the value of $n$ for which the $n^{th}$ term of $a_n = n^2 - 5n + 6$ is zero.

Calculate $a_3$ if $a_n = \frac{n^3 + n}{n^2 + 1}$.

Find the $n^{th}$ term for the sequence $5, 8, 11, 14, \dots$.

Determine the $10^{th}$ term of the sequence $a_n = 3n - 3$.

What is the first term ($a_1$) of the sequence $a_n = \frac{n}{n+1}$?

If the rule is $a_n = 4n + 4$, find the $4^{th}$ term.

Find the $2^{nd}$ term of the sequence $a_n = 10^n$.

Calculate the $3^{rd}$ term of the sequence defined by $a_n = \frac{3^n}{n}$.

What is the explicit rule for the sequence $9, 19, 29, 39, \dots$?

Determine the $6^{th}$ term of the sequence $a_n = \frac{2}{3}n + 1$.

Identify the explicit rule for $4, 10, 18, 28, 40, \dots$.

If $a_n = n^2 + k$, and $a_3 = 14$, what is $a_5$?

Find the $4^{th}$ term of the sequence $a_n = \frac{(n+1)!}{(n-1)!}$.

Given $a_1 = 4$ and $a_n = 2a_{n-1} - 2$, find $a_2$.

If $a_1 = 10$ and $a_n = a_{n-1} - 10$, find $a_3$.

What is $a_2$ for the rule $a_n = a_{n-1} + 13$ with $a_1 = 7$?

If $a_1 = 3$ and $a_n = (a_{n-1})^2$, find $a_3$.

For the sequence $a_1 = 0$ and $a_n = (a_{n-1} + 1)^2 - 1$, find $a_3$.

Calculate the 3rd term for $a_1 = 9$ and $a_n = 2\sqrt{a_{n-1}}$.

If $a_1 = 1, a_2 = 2$ and $a_n = 2a_{n-1} + a_{n-2}$, find $a_4$.

In a sequence where $a_1 = 1$ and $a_n = a_{n-1} + \frac{1}{n(n-1)}$ for $n > 1$, calculate $a_{100}$.

If $a_1 = 3$ and $a_n = 2a_{n-1} + n$, find the value of $a_4$.

Let $a_1 = 1$ and $a_n = a_{n-1} + 2^{n-1}$. What is $a_6$?

If the first term is $0$ and the common difference is $7$, what is the $3^{rd}$ term?

Find the next term: $-2, -4, -6, -8, \dots$.

What is the first term of the sequence $x-y, x, x+y, \dots$?

What term comes after $21, 18, 15, 12, \dots$?

In the sequence $8, 14, 20, 26, \dots$, which term is exactly $72$ more than the $5^{th}$ term?

Find the common difference of the AP: $-1.1, -3.1, -5.1, -7.1, \dots$.

The first three terms of an AP are $3y-1, 3y+5$, and $5y+1$. Find the value of $y$.

Find the sum of first $20$ terms of an AP whose $n^{th}$ term is given by $a_n = 5n - 1$.

The sum of the first $n$ terms of two APs are in the ratio $(7n+1) : (4n+27)$. Find the ratio of their $11^{th}$ terms.

If the sum of the first $n$ terms of an AP is $S_n$ and $S_{2n} = 3S_n$, find the ratio $S_{3n} / S_n$.

Find the value of $\sum_{i=1}^{9} i$.

If the sum of the first $n$ natural numbers is 55, what is $n$?

What is the sum of the first 49 natural numbers?

Find the sum of the first 44 natural numbers.

Calculate the sum of all natural numbers from $1$ to $60$, excluding those that are multiples of $10$ (i.e., $10, 20, 30, 40, 50,$ and $60$).

What is the smallest natural number $n$ such that the sum of the first $n$ natural numbers is greater than $500$?

If the sum of the first $k$ natural numbers is $435$, what is the value of $k^2$?

Let $S_n$ denote the sum of the first $n$ natural numbers. If $n$ is a natural number such that $S_{n+2} = 2S_n + 1$, then find the value of $S_{2n}$.

What is the value of $(1+2+3+\dots+100) - (1+2+3+\dots+99)$?

Find $n$ if $1 + 2 + 3 + \dots + n = \frac{1}{2}(n^2 + n)$.

In a GP, if $a = 2$ and $a_2 = 8$, what is the common ratio?

Find the next term in the sequence $0.5, 1.5, 4.5, \dots$.

What is the common ratio of the sequence $-2, 4, -8, 16, \dots$?

Find the $2^{nd}$ term of a GP if $a = 15$ and $r = \frac{1}{3}$.

What is the common ratio of the sequence $7, 14, 28, 56, \dots$?

In a GP, if $a=1$ and $a_4=64$, what is the common ratio?

Find the value of $y$ if $3, y, 27$ are in GP ($y > 0$).

If the sum of $n$ terms of a GP is $S_n$, find $S_{3n} : (S_{2n} - S_n)$.

Find the sum of $n$ terms of the series $5 + 55 + 555 + \dots$

In a GP, if the $3^{rd}$ term is $S$, then the product of the first $5$ terms is $S^k$. What is the value of $k$?