General term of a sequence

Given the sequence whose nth term is $a_n = \frac{2n-1}{n+3}$, calculate its first five terms.

Check mathematically whether the number $47$ belongs to the sequence with nth term $c_n = 3n + 2$. If so, state its position in the sequence.

Given the sequence whose general term is $a_n = \frac{n^2+1}{n}$, find the expression for the general term of the difference sequence defined as $d_n = a_{n+1} - a_n$ and simplify the result as much as possible.

Write down the nth term $f_n$ of the following sequence of fractions:

$\frac{1}{2}, \frac{4}{5}, \frac{9}{10}, \frac{16}{17}, \dots$

Find the nth term $a_n$ of the following sequence by combining the patterns governing its numerator and denominator independently:

$\frac{2}{3}, \frac{5}{9}, \frac{8}{27}, \frac{11}{81}, \dots$

Write down the nth term of a sequence whose terms are consecutive integers with alternating signs:

$-1, 2, -3, 4, -5, \dots$

Find a single explicit general term $a_n$ for the sequence $1, 4, 3, 8, 5, 12, 7, 16, \dots$ using the expression $(-1)^n$ to unify the differing behaviour shown by the even and odd terms.

Given the recurrence relation $a_1 = 2$ and $a_n = 3a_{n-1} - 1$ for $n \ge 2$, calculate the first four terms of the sequence.

A recurrence sequence is defined by the initial conditions $x_1 = 1$, $x_2 = 2$ and the recurrence relation $x_n = x_{n-1} + 2x_{n-2}$ for all $n \ge 3$. Calculate the terms of the sequence up to $x_6$ and deduce an explicit formula for the general term $x_n$ in terms of powers of $2$.

The sequence $a_n$ is defined by the recurrence relation $a_1 = 3$ and $a_n = 2a_{n-1} + 1$ for $n \ge 2$. To find its explicit general term, define a new auxiliary sequence $b_n = a_n + 1$. Prove that $b_n$ is a geometric progression, find its general term $b_n$ and, from it, deduce the general term of $a_n$.

A population of microscopic algae in a controlled lake triples its covered surface area each week. If in the first week (week $n=1$) they cover a surface area of $2 \text{ m}^2$:

1. a) Write down the nth term $S_n$ representing the covered surface area after $n$ weeks.
2. b) Calculate how many full weeks must pass for the covered surface area to exceed $1000 \text{ m}^2$ for the first time.

The market value of a new car is $24\,000$ €. It is estimated that each year it loses $15\,\%$ of its value compared to the immediately preceding year. Find the nth term $V_n$ expressing the value of the car after $n$ years (considering $n=0$ to be the time of purchase), and determine its estimated value after $5$ years.

An amateur athlete is preparing for a half marathon. On the first day of training, he runs $5$ km, and decides to increase the daily distance by exactly $500$ metres compared to the previous day. Express the general term $d_n$ for the distance in kilometres run on day $n$. On which day of his training will he run exactly $15$ km?