Exercises: Arithmetic progressions

Write down the general term and calculate the twentieth term of the following arithmetic progression: $7, 11, 15, 19, \dots$

In an arithmetic progression, the first term is $a_1 = -5$ and the common difference is $d = \frac{3}{2}$. Calculate the value of the term in position $40$.

Given the sequence defined by the general term $b_n = 5 - 3n$, prove that it is an arithmetic progression and determine its common difference.

Calculate the sum of the first $30$ terms of an arithmetic progression, knowing that the first term is $a_1 = 4$ and the thirtieth term is $a_{30} = 91$.

The eighth term of an arithmetic progression is $32$ and its common difference is $d = 4$. Find the first term $a_1$ and the expression for its general term $a_n$.

Determine whether the number $102$ belongs to the arithmetic progression whose general term is $c_n = 4n - 6$. If so, state its position in the sequence.

Calculate the sum of the first $50$ positive multiples of $6$ (that is, the sum of the sequence $6, 12, 18, \dots$).

The first three terms of an arithmetic progression are $x$, $x+4$ and $2x-1$. Calculate the numerical value of $x$ and write down these first three terms.

Seán has to solve the following exercise: “Interpolate $3$ arithmetic means between the numbers $8$ and $28$”. In his exam, he writes the following:

“Since $3$ numbers need to be interpolated, the difference between consecutive terms is calculated by dividing the total distance by $3$:

Therefore, the requested terms are: $8 + 6.67 = 14.67$; $14.67 + 6.67 = 21.34$; and $21.34 + 6.67 = 28.01$”.

1. a) Identify the conceptual error made by Seán and explain, using the idea of “intervals” or “steps” between terms, why his denominator is incorrect.
2. b) Correct the working by calculating the exact value of the difference $d$ and write down the $3$ correct arithmetic means.

Cian is training for a $10\text{ km}$ community race in his town. His coach has scheduled a plan of $20$ consecutive days of running: on the first day he will run $2\text{ km}$ and, each successive day, he will increase the distance by exactly $300\text{ m}$ ($0.3\text{ km}$) compared to the previous day.

1. a) Before applying any formula, make a reasoned estimation of the total volume: if he ran every day at the pace of the first day, he would accumulate $40\text{ km}$, and if he ran every day the distance of the last day, he would accumulate significantly more. Mentally estimate between which two round values (multiples of $10$) the total sum will lie and state whether you consider it plausible that it exceeds $100\text{ km}$.
2. b) Model the daily distance using an arithmetic progression, determine the distance he will run on day $20$, and use the summation formula to calculate the exact total number of kilometres run during the $20$ days.
3. c) Check whether your initial estimation was accurate and reflect on whether the proposed daily increase is suitable for avoiding injuries.

A construction company establishes a monthly bonus scheme to motivate its employees. In the first month, an employee receives $50$ €, in the second month $65$ €, in the third month $80$ €, and so on. How much money in total will the employee have received in bonuses after $2$ full years of work?

A cyclist begins her training plan by riding $12$ km on the first day. To improve her endurance, each day she increases the distance of her ride by $1.5$ km compared to the previous day. On which day of her training will she ride exactly $42$ km? How many total kilometres will she have ridden, adding up all the days up to and including that day?

In an arithmetic progression, it is known that the fourth term is $15$ and the tenth term is $39$. Set up a system of equations to calculate the common difference $d$ and the first term $a_1$. Subsequently, find the sum of the first $20$ terms of this progression.

Interpolate five arithmetic means between the numbers $-8$ and $22$. Write down the complete resulting arithmetic progression, stating its common difference.

A theatre has its seats arranged in a fan shape. The first row has $20$ seats, the second $24$, the third $28$, and so on in arithmetic progression. If the theatre has a total of $15$ rows, how many seats are in the last row? What is the total capacity (total number of seats) of the theatre?

A freely falling object on a fictional planet travels $4.9$ m in the first second. In each subsequent second, the distance travelled during that second increases by $9.8$ m compared to the distance travelled in the previous second. What distance does the object travel exclusively during the tenth second of its fall?

In an arithmetic progression, the first term is $a_1 = 3$ and the common difference is $d = 2$. If the sum of the first $n$ terms is $S_n = 360$, set up the corresponding quadratic equation and determine the number of terms $n$ that have been added.

To reforest a wood after a fire, a forestry team plants trees progressively. In the first week, they manage to plant $120$ trees, but due to accumulated fatigue and less available accessible land, each week they plant $5$ fewer trees than the previous week. How many weeks will pass until the team has planted an exact total of $1470$ trees? (Note: Analyse the possible solutions mathematically and justify which one makes sense in the context of the problem).

Following the forest fires in the region of the Sierra de la Culebra (Zamora), an environmental group organizes Sunday reforestation sessions with native species. During the first week, they manage to plant $140$ saplings. In the following weeks, thanks to the continuous arrival of new volunteers, they manage to plant $25$ more saplings each week than in the previous one.

1. a) Write down the expression for the general term $a_n$ that determines how many saplings are planted in week $n$.
2. b) The goal of the municipal campaign is to reach an accumulated total of at least $3\,500$ saplings planted. Set up the quadratic equation that allows you to find out in which week this goal will be achieved.
3. c) Solve the equation, state how many full weeks of work will be necessary, and interpret the algebraic meaning of the negative solution, discarding it with a contextual argument.

In an arithmetic progression, it is known that the term in the fifth position is $a_5 = 19$ and the term in the eighteenth position is $a_{18} = 71$. To determine the common difference $d$ and the first term $a_1$, two students propose different approaches:

• Aoife sets up a system of two linear equations with two unknowns from the standard formula:

• Seán proposes relating both terms directly by counting the number of steps between them:

1. a) Solve the problem following both methods and verify that identical values for $d$ and $a_1$ are obtained.
2. b) Justify which method is more efficient in this case (fewer operational steps and lower risk of algebraic error).
3. c) Using the method you consider most advantageous, argue formally whether the number $250$ can be a term in this sequence or if it is impossible for it to belong to it.

In an arithmetic progression with common difference $d = 3$, the first term is $a_1 = -10$. Calculate the sum of the terms between position $20$ and position $40$, inclusive.

The interior angles of a convex hexagon are in arithmetic progression. Knowing that the largest angle measures $140^\circ$, calculate the measures of the other five interior angles of the hexagon. (Remember that the sum of the interior angles of an $n$-sided polygon is $180^\circ \times (n-2)$).

Determine the values of the parameter $k$ such that the algebraic expressions $k^2 - 2$, $3k + 1$ and $k^2 + 4k$ are three consecutive terms of an arithmetic progression. For each value of $k$ found, write down the corresponding three terms.

The sum of the first $n$ terms of a sequence is given by the expression $S_n = 2n^2 + 5n$. Find the mathematical expression for the general term $a_n$ and prove rigorously that this sequence is an arithmetic progression.

In an industrial warehouse, cylindrical steel pipes are stacked such that each row has exactly one pipe fewer than the row immediately below it on which it rests. If the top row (the peak) has just $1$ pipe and there are $210$ stacked pipes in total, how many rows of pipes make up the stack? How many pipes are in the bottom row resting on the ground?

Find three numbers in arithmetic progression knowing that their sum is $24$ and that the sum of their squares is $242$.

Prove algebraically that if the numbers $a^2$, $b^2$ and $c^2$ are in arithmetic progression, then the numbers $\frac{1}{b+c}$, $\frac{1}{a+c}$ and $\frac{1}{a+b}$ also form an arithmetic progression.

A company purchases an industrial machine for $32000$ € and reaches an agreement to pay for it in monthly instalments that form an arithmetic progression. If it is known that the first instalment to be paid is $500$ € and the last instalment will be $2700$ €, in how many months will the company settle its debt? What is the exact monthly increase in the instalment compared to the previous month?

Instead of giving you the terms of an arithmetic progression to calculate its sum, only the polynomial function that provides the accumulated sum of its first $n$ terms is known:

1. a) Deduce with justification the value of the first term $a_1$ by appropriately evaluating $S_1$.
2. b) From the conceptual principle $a_n = S_n - S_{n-1}$ (for all $n \ge 2$), obtain the simplified expression for the general term $a_n$ in terms of $n$.
3. c) Formally prove that the resulting sequence is an arithmetic progression by calculating the difference $a_n - a_{n-1}$ and verifying that it is constant.
4. d) Now consider the complementary inverse problem: design the algebraic expression for the sum $S_n$ corresponding to an arithmetic progression whose first term is $a_1 = 10$ and whose common difference is $d = -3$. Check that your expression is correct by calculating $S_4$ with it and comparing it with the manual sum of the first four terms.

A municipal auditorium in Galway plans to renovate the seats in its stalls with a tiered trapezoidal layout. Technical regulations and the space design impose the following conditions:

1. The first row (closest to the stage) will have $a_1 = 15$ seats.

2. Each subsequent row will have exactly $2$ more seats than the previous one ($d = 2$).

3. For safety and the width of evacuation routes, the total capacity of the venue cannot exceed $650$ spectators under any circumstances, but the city council requires maximizing the capacity without exceeding that limit.

4. The physical depth of the hall does not allow more than $25$ rows to be built.

1. a) Algebraically model the accumulated capacity $S_n$ as a function of the number of rows $n$.
2. b) Set up the capacity inequality $S_n \le 650$, solve the associated quadratic equation, and determine, justifying your choice within the set of natural numbers, the exact number of rows $n$ that must be installed to achieve the maximum permitted capacity.
3. c) State how many seats there will be in the last installed row and what the final total capacity of the auditorium amounts to.
4. d) If, due to technical visibility reasons, it is ultimately decided to remove the first $5$ full rows (with the audience starting to sit from what used to be row $6$) while keeping the same last row calculated in the previous section, how many usable seats will remain in the auditorium? Solve this by applying the properties of the sum of arithmetic progressions without calculating row by row.