The nth term of a sequence

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

Find the expression for the nth term of the following arithmetic sequence:

$5, 9, 13, 17, 21, \dots$

Find the expression for the nth term of the following geometric sequence:

$3, -6, 12, -24, 48, \dots$

If the nth term of a sequence is $b_n = n^2 - 5n + 6$, find the exact value of the twentieth term ($b_{20}$).

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.

Observe the following sequence of figures made up of adjacent hexagons sharing one or more sides. Figure 1 is made of 6 matchsticks, Figure 2 of 11 matchsticks and Figure 3 of 16 matchsticks. Write down the nth term for the number of matchsticks $P_n$ needed to build Figure $n$.

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.

Find the general term of the sequence $0, 3, 8, 15, 24, \dots$ and use it to calculate the exact value of the fiftieth term ($a_{50}$).

Consider the sequence defined by the general term $a_n = \frac{5n^2 - 3}{2n^2 + 7n}$.

1. a) Before performing any exact calculation, make a reasoned estimate of the value that the terms of the sequence will approach when the position $n$ is extremely large (for example, for $n = 10\,000$). Justify which terms of the fraction you focus on and why the others barely influence the result.
2. b) Check your estimate by calculating the exact value (as an irreducible fraction) and the decimal approximation to three decimal places for $n = 1$, $n = 10$, and $n = 100$. Do these numerical values support your initial prediction?
3. c) Set up the equation needed to find out whether the number $2.5$ is exactly a term of the sequence. Solve it and explain why, despite the estimate in part a, it is impossible for $2.5$ to be part of it.

Prove mathematically, by studying the algebraic sign of the difference $a_{n+1} - a_n$, whether the sequence $a_n = \frac{3n-1}{n+2}$ is strictly increasing or strictly decreasing.

In a computer network, the total number of possible direct connections between $n$ computers (if all are physically connected to each other) forms a sequence $C_n$. Knowing that $C_1 = 0$, $C_2 = 1$, $C_3 = 3$, and $C_4 = 6$, determine the general term $C_n$. From it, deduce how many computers make up a network that has exactly $1225$ direct connections.

Study the boundedness of the sequence $a_n = \frac{5n+2}{2n+1}$. To do so, prove algebraically that the sequence is bounded below by $\frac{7}{3}$ and bounded above by $\frac{5}{2}$.

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:

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

Cian wants to find the general term of the sequence of fractions:

On his exam, he writes the following justification:

"The numerator increases by 2 starting at 3, so it is $2n+1$. The denominator consists of the squares of $2, 3, 4, 5\ldots$, so it is $(n+1)^2$. Since the signs alternate between positive and negative, I multiply by $(-1)^n$. Therefore, the general term is $a_n = (-1)^n \frac{2n+1}{(n+1)^2}$".

1. a) Calculate the first two terms ($a_1$ and $a_2$) using Cian's formula and precisely identify the mistake he made.
2. b) Explain how the exponent of the power with base $-1$ must be modified to correct the sign, and write the correct expression for the general term.
3. c) Cian claims that the value $-\frac{21}{121}$ belongs to this sequence and that it occupies an even position. Use the corrected general term to reason whether his statement is true or false.

Design the algebraic expression for the general term $a_n$ of a sequence that simultaneously satisfies these four conditions:

1. The first term ($n=1$) is negative and the sign alternates between each term and the next.

2. All its terms are fractions whose absolute value is strictly less than 1.

3. As the index $n$ increases, the absolute value of the terms progressively approaches 1 (that is, the numerator and denominator get closer in absolute value).

4. The number $-\frac{5}{6}$ belongs to the sequence.

Once your formula is defined:

1. a) Write its first three terms ($a_1, a_2, a_3$) to check that conditions 1, 2, and 3 are satisfied.
2. b) Set up and solve the equation that allows you to find out which position $n$ is occupied by the number $-\frac{5}{6}$, justifying why the result guarantees that this value belongs to the sequence.

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?

A rubber ball is dropped from an initial height of $10$ metres. On each bounce against the ground, the ball reaches a height equal to $\frac{3}{4}$ of the maximum height reached on the immediately preceding bounce. Write down the general term $h_n$ for the height reached after the $n$th bounce. Using logarithms, determine after how many bounces the maximum height will be less than $1$ metre for the first time.