Exercises: Arithmetic progressions

State which of the following sequences are arithmetic progressions and, where applicable, give the common difference:

1. a) $4, 7, 10, 13, \ldots$

1. b) $2, 4, 8, 16, \ldots$

1. c) $20, 17, 14, 11, \ldots$

1. d) $1, 1, 1, 1, \ldots$

1. e) $3, 7, 12, 18, \ldots$

An arithmetic progression has first term $a_1 = 5$ and common difference $d = 3$.

1. a) Write its first five terms.

1. b) Write the general term $a_n$.

1. c) Calculate $a_{10}$.

In an arithmetic progression, $a_1 = 2$ and $a_5 = 14$.

1. a) Calculate the common difference.

1. b) Write the general term.

1. c) What position does the term $26$ occupy?

Consider the arithmetic progression $3, 7, 11, \ldots$

1. a) Calculate the sum of its first $10$ terms.

1. b) Check the result by first calculating $a_{10}$ and applying the sum formula.

Aoife wants to save for her end-of-year school trip. She starts by saving $€25$ the first month and decides to increase her contribution by $€5$ each month for a full year (12 months). To find out how much money she will gather, she writes in her copy:

$$a_{12} = 25 + (12 - 1)\times 5 = 25 + 55 = €80$$

And concludes: "Great! At the end of the year I will have $€80$ saved for the trip".

1. a) Identify and explain in detail the conceptual error Aoife makes when interpreting the result of $a_{12}$. What does that figure of $€80$ actually represent?
2. b) Correct the calculation by applying the appropriate formula to determine the total amount Aoife will have accumulated in her piggy bank at the end of the 12 months.
3. c) If the trip costs $€600$, reason whether the saved money will be enough, or how much will be missing or left over.

A student opens an electronic wallet account to pay for their bus travel during the school year. The first month they deposit $\text{€}\ 40$ and each month they add $\text{€}\ 10$ more than the previous month, keeping this plan over 10 months.

1. a) Before performing detailed calculations, make a reasoned estimate of whether the total saved at the end of the 10 months will be closer to $\text{€}\ 400$, $\text{€}\ 850$, or $\text{€}\ 1,500$. Justify your estimate by considering a representative monthly payment (the average payment between the first and the last month).
2. b) Model the situation using an arithmetic progression: identify the first term $a_1$, the common difference $d$, and calculate the exact amount contributed in the tenth month ($a_{10}$).
3. c) Calculate the total accumulated sum $S_{10}$ and assess whether your initial estimate was accurate or not, explaining what accounts for the agreement or deviation.

A savings account at $3 \%$ simple annual interest starts with an initial capital of $\text{€}\ 2,000$ that is not withdrawn: each year the same interest on the initial capital is added.

1. a) Why do the capital amounts year after year form an arithmetic progression? State its first term and common difference.

1. b) Write the general term for the capital accumulated after $n$ years.

1. c) How much capital will there be after $6$ years? And after $15$ years?

For the end-of-year trip, a student decides to save using a staircase system: €10 in the first week and, each subsequent week, €5 more than the previous one, as shown in the figure.

1. a) How much will she save in week $12$?

1. b) How much will she have accumulated in total after those $12$ weeks?

1. c) The trip costs €650. Will saving for $16$ weeks be enough? Justify your answer.

Insert four arithmetic means between $3$ and $18$ (that is, construct an arithmetic progression of six terms whose extremes are $3$ and $18$).

1. a) Calculate the common difference of the progression.

1. b) Write the six terms.

1. c) Check that the sum of the six terms matches the formula $S_n = \frac{(a_1 + a_n) \times n}{2}$.

A park amphitheatre has $22$ seats in the first row, and each subsequent row has $4$ seats more than the previous one.

1. a) How many seats are in row $8$?

1. b) How many seats are there in total in the first $8$ rows?

1. c) A total of $600$ seats are needed for a concert. What is the minimum number of complete rows required?

The figure uses bars to represent the first six terms of an arithmetic progression.

1. a) Read the first term $a_1$ and the common difference $d$ from the graph, and explain why the heights of the bars lie on a straight line.

1. b) Write the general term $a_n$.

1. c) Calculate $a_{20}$ and the sum of the first $20$ terms.

Consider the arithmetic progressions $a_n$, with $a_1 = 7$ and common difference $2$, and $b_n$, with $b_1 = 40$ and common difference $-1$.

1. a) Write the general term for each one.

1. b) Is there any value of $n$ for which $a_n = b_n$? Calculate it.

1. c) Interpret the result: describe in words how each progression evolves and what the common term means.

A credit union in Munster designs a microcredit program of $€3,960$ for entrepreneurs to be repaid in 12 decreasing monthly installments structured in an arithmetic progression. Due to the institution's social policy, two conditions are established: the sum of the 12 installments must exactly amortize the total amount owed ($€3,960$) and the first installment must be exactly twice the last one ($a_1 = 2\times a_{12}$).

1. a) Using the formula for the sum of an arithmetic progression, set up an equation relating $a_1$ and $a_{12}$ to the total principal, and deduce the exact amount of the first installment ($a_1$) and the last one ($a_{12}$).
2. b) Using the values found, calculate the common difference $d$ of the progression and write the general term $a_n$ that determines the payment for any month $n$.
3. c) Interpret the sign of the common difference $d$ in this financial context and provide a reasoned justification for why a decreasing installment scheme can be advantageous for a start-up business compared to a traditional constant installment.

To finance the tuition for a university master's degree amounting to €5,400, a graduate in Ireland establishes a monthly savings plan over 2 years (24 months). The first month she deposits €110 and commits to increasing her monthly contribution by a constant fixed amount $d$ each subsequent month.

1. a) Model the savings plan using an arithmetic progression and obtain the simplified algebraic expression for the total accumulated capital $S_{24}$ in terms of the common difference $d$.
2. b) Calculate the exact monthly increase $d$ she must apply to gather exactly €5,400 at the end of month 24. With that value of $d$, determine what her contribution will be in the last month ($a_{24}$).
3. c) After 12 months (exactly halfway through the planned time), will she have managed to save half of the tuition cost (€2,700)? Calculate $S_{12}$ and reason quantitatively why, in plans with increasing contributions, the accumulated savings do not grow directly proportional to the elapsed time.

How many terms of the arithmetic progression $5, 9, 13, \ldots$ must be added to obtain $377$?

1. a) Write the general term $a_n$ and the expression for the sum $S_n$ of the first $n$ terms.

1. b) Set up the equation $S_n = 377$ and solve it. Why must one of the solutions to the quadratic equation be discarded?

The lengths of the three sides of a right-angled triangle are in arithmetic progression with common difference $d > 0$.

1. a) Express the three sides in terms of the middle side $b$ and the common difference $d$, and apply Pythagoras' theorem.

1. b) Show that the sides are necessarily proportional to $3$, $4$ and $5$.

1. c) Give a concrete example of a right-angled triangle whose sides form an arithmetic progression with common difference $7$.

State with reasons whether the following statements about arithmetic progressions are true or false, proving them or giving a counterexample:

1. a) "Three numbers $a$, $b$ and $c$ are in arithmetic progression if and only if $b = \frac{a + c}{2}$."

1. b) "In an arithmetic progression, the sum of two terms equidistant from the extremes is always the same: $a_1 + a_n = a_2 + a_{n - 1} = a_3 + a_{n - 2} = \ldots$"

1. c) "If the same quantity is added to all the terms of an arithmetic progression, the common difference changes."

A father offers his daughter two monthly allowance plans:

• Plan A: $\text{€}\ 20$ in the first month, increasing by $\text{€}\ 2$ each month.

• Plan B: $\text{€}\ 8$ in the first month, increasing by $\text{€}\ 3$ each month.

1. a) Write the general term for the allowance under each plan in month $n$.

1. b) In which month does the monthly allowance of both plans coincide for the first time?

1. c) In which month does the cumulative total of plan B exceed that of plan A for the first time? Justify your answer using the sum formulas and reasoned trial and error.

An investor deposits an initial principal $C_0$ in a bank deposit bearing simple annual interest at a rate of $4\text{ \%}$ ($r = 0.04$). They want to find out how many full years must pass for the total accumulated capital to reach at least three times the initial principal invested ($C_n \ge 3\times C_0$). Two students propose different strategies:

• Seán's method (general term): He models the evolution as an arithmetic progression whose first term is the capital at the end of the first year, $a_1 = C_0(1 + 0.04) = 1.04\times C_0$, and whose annual common difference is the fixed interest generated, $d = 0.04\times C_0$. He sets up $a_n = a_1 + (n - 1)\times d \ge 3\times C_0$ and solves for $n$.

• Aoife's method (accumulated interest): She reasons that, to triple the total capital, the net interest generated must equal twice the initial principal ($I = 2\times C_0$). She applies the simple interest formula $I = C_0\times r\times n$, sets up $C_0\times 0.04\times n \ge 2\times C_0$, and solves for $n$.

1. a) Solve the problem applying both methods step by step. Do you obtain the same result? Does the time obtained depend on the specific amount of the initial principal $C_0$?
2. b) Critically compare both approaches: which one is more direct and reduces the risk of making time-lag errors in the index $n$?
3. c) Generalize the result: if the annual simple interest is $r\text{ \%}$ and the goal is to multiply the initial principal by a factor $k$ ($k > 1$), deduce the simplified formula that directly provides the required years $n$ in terms of $r$ and $k$. Explain which method allowed you to deduce it more clearly.

Two coworkers, Seán and Aoife, take out 10-year (120-month) savings plans with a financial institution:

• Plan A (Seán): Contributes €100 the first month and increases his contribution by €2 each month ($d_A = €2$).

• Plan B (Aoife): Starts with a lower contribution, €40 the first month, but increases her monthly payment at a faster pace, by €3 each month ($d_B = €3$).

1. a) Determine in which specific month Aoife's monthly contribution exceeds Seán's for the first time ($b_n > a_n$). Does that mean that in that same month Aoife will have already accumulated more total money than Seán? Justify your answer without calculating the sums yet.
2. b) Set up the expressions for the accumulated sums $S_n^A$ and $S_n^B$ and solve the equation $S_n^B = S_n^A$. Interpret the two solutions yielded by the quadratic equation: what financial meaning does the solution $n = 0$ have, and in which exact month would Aoife tie with Seán in total accumulated money if the plan continued indefinitely?
3. c) Knowing that the plan matures after 120 months, calculate the final accumulated capital for each of them. Explain the apparent contradiction of why Seán finishes with more money if Aoife has been paying higher monthly installments for practically half of the plan's duration.