Arithmetic sequences and series

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?

A savings account at $3 \%$ simple annual interest starts with an initial capital of $\text{€}\ 2,000$ that is not withdrawn: each year the same interest on the initial capital is added.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

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.

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.

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.

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.

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?

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 $S_n = \frac{(a_1 + a_n) \times n}{2}$.

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."

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 $7$.

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?

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.

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.