The nth Term of a Sequence

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

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

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

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

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

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

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

Write the general term $f_n$ of the following sequence of fractions:

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

Find the general 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 the general term of a sequence whose terms are consecutive integers but 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 behavior shown by the even and odd terms.

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 ($a_{50}$).

In a computer network, the total number of possible direct connections between $n$ computers (if all are physically connected to all others) 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 this, deduce how many computers make up a network that has exactly $1225$ direct connections.

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 this, deduce the general term of $a_n$.

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

A geometric fractal is constructed from an equilateral triangle of area $A$. In each step, the midpoints of the sides of the unshaded triangles are joined to form 4 smaller triangles, and only the central one is shaded. Find the general term $T_n$ expressing the number of new shaded triangles added in step $n$, and the general term $A_n$ for the area of just one of those new triangles added in step $n$.

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

1. a) Write the general term $S_n$ representing the covered area after $n$ weeks.
2. b) Calculate how many full weeks must pass for the covered 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 general term $V_n$ expressing the value of the car after $n$ years (considering $n=0$ as the time of purchase), and determine its estimated value after $5$ years.

A rubber ball is dropped from an initial height of $10$ feet. 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 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 $1$ foot for the first time.

An amateur athlete is preparing for a half marathon. On the first day of training they run $5$ miles, and decide to increase the daily distance by exactly $500$ feet compared to the previous day. Express the general term $d_n$ for the distance in miles run on day $n$. On which day of their training will they run exactly $15$ miles?

$\$5\,000$ are invested in an investment fund offering a compound annual interest of $4\,\%$. However, at the end of each year and after applying the interest, the entity deducts a fixed $\$50$ as management fees. Write the recurrence relation for the capital $C_n$ accumulated at the end of year $n$ (where $C_0 = 5000$), and calculate the capital in the first three years ($C_1, C_2, C_3$).

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 fully.

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.

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