Exercises: Arithmetic progressions

Elena 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 notebook:

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

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 Elena 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 Elena 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 a digital wallet account to pay for their bus fares during the school year. The first month they deposit $\$40$ and each month they add $\$10$ 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 $\$400$, $\$850$, or $\$1,500$. 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 sequence: 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A savings account at $3 \%$ simple annual interest starts with an initial capital of $\$2,000$ that is not withdrawn: each year the same interest on the initial capital is added.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 sequence? 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.

A park amphitheater 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?

A credit union in Arizona 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 college graduate in the U.S. 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.

A father offers his daughter two monthly allowance plans:

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

• Plan B: $\$8$ in the first month, increasing by $\$3$ 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:

• Ethan'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 $n$.

• Mia'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 off-by-one 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, Carlos and Lucía, take out 10-year (120-month) savings plans with a financial institution:

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

• Plan B (Lucía): 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 Lucía's monthly contribution exceeds Carlos's for the first time ($b_n > a_n$). Does that mean that in that same month Lucía will have already accumulated more total money than Carlos? 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 Lucía tie with Carlos 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 Carlos finishes with more money if Lucía has been paying higher monthly installments for practically half of the plan's duration.