Exercises: nth term

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:

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

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

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

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.

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

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

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

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.

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.

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.

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

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

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 $n$.

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.

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.

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 $2$.

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?

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.

An amateur athlete in Dublin is preparing for a 10 km road race in Phoenix Park following a 16-week training plan. In the first week ($n=1$), he completes a long run of $4 \text{ km}$. The coach establishes a linear base progression that adds $0.5 \text{ km}$ per week (that is: $4 \text{ km}$ in week 1, $4.5 \text{ km}$ in week 2, $5 \text{ km}$ in week 3, etc.). However, to assimilate muscular effort, he introduces a periodic deload rule:

• If week $n$ is odd, the athlete runs the full distance set by the base progression.

• If week $n$ is even, the athlete reduces by $1 \text{ km}$ the run that would correspond to him according to the base progression.

1. a) Model the actual distance $T_n$ (in kilometers) of the run in week $n$ using a single unified explicit formula, using a power with base $-1$ to alternate the adjustment between even and odd weeks.
2. b) Determine algebraically in which week he will exceed $9 \text{ km}$ in his long run for the first time.
3. c) How many kilometers will he have to run in week $n=16$, the last loading week before the race?

A geometric fractal is constructed starting from an equilateral triangle of area $A$. At 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 at step $n$, and the general term $A_n$ for the area of a single one of those new triangles added at step $n$.

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.

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.

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.

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.

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 $a_n$.

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 $\frac{5}{2}$.

$5\,000$ € is invested in an investment fund that offers an annual compound interest rate of $4\,\%$. However, at the end of each year and after applying the interest, the institution deducts a fixed $50$ € for management fees. Write down 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 ($C_1, C_2, C_3$).

A sequence is defined recursively by $a_1 = 2$ and $a_{n+1} = \frac{a_n}{1 + 3a_n}$ for all $n \geq 1$. Two students must solve two tasks: calculate the term $a_3$ and determine whether the value $\frac{1}{152}$ belongs to the sequence.

• Aoife proposes applying the recurrence successively term by term until both questions are answered.

• Seán proposes calculating the first four terms, studying the auxiliary sequence of their reciprocals $b_n = \frac{1}{a_n}$ to find a general rule, deducing the explicit expression for $a_n$ in terms of $n$, and working from there.

1. a) Develop Seán's method: find $a_1, a_2, a_3, a_4$, write the corresponding values of $b_n$, deduce with reasoning the formula for the explicit general term $a_n$, and check that it matches the values found.
2. b) Compare both procedures for the calculation of $a_3$ alone. Which one is more economical in time and operations for that particular case? Justify your answer.
3. c) Use the explicit general term to determine with mathematical rigor whether the number $\frac{1}{152}$ belongs to the sequence. Then, explain why Aoife's recurrence method would be inefficient or unfeasible to answer this question with certainty.

Consider the sequence whose first terms are:

1. a) Analyze the pattern followed by the terms in odd positions ($n=1, 3, 5, \ldots$) and the one followed by those in even positions ($n=2, 4, 6, \ldots$). Write the rule that determines each group of terms in terms of the global index $n$.
2. b) To express the sequence in a single closed formula without resorting to a piecewise-defined function, "algebraic switches" built with powers of base $-1$ are used. Justify what values the expressions $\frac{1 - (-1)^n}{2}$ and $\frac{1 + (-1)^n}{2}$ take depending on the parity of $n$, and use them to deduce a unified and simplified formula for the general term $a_n$.
3. c) Investigate algebraically whether the number 100 is part of this sequence. Set up the corresponding equations and explain how many times and in what exact position or positions this value appears.