Exercises: Geometric progressions

A young content creator on a social network observes that the number of views on his videos is increasing steadily. On the first day he obtained $200$ views, on the second $600$, on the third $1800$, and so on, following a geometric progression.

1. a) Determine the common ratio $r$ of the progression.
2. b) Calculate how many views he will receive on the sixth day if this trend continues.

To buy an electric bicycle, a student decides to do the "weekly savings challenge": she saves money every week such that each deposit is double the previous one. The first week she saves $2$ €, the second $4$ €, the third $8$ €, and so on.

1. a) Write the general term $a_n$ of this progression.
2. b) Use the sum formula for a geometric progression to find out how much money she will have accumulated in total after $8$ weeks.

A grandfather opens a savings account for his newborn grandson with an initial capital of $2000$ €. The bank offers simple annual interest of $2,5 \%$.

1. a) Calculate the interest generated by the account after $10$ years.
2. b) What will the total capital accumulated amount to after that period?

Due to inflation, rent prices in a youth district of the city have been rising by $4 \%$ each year cumulatively. If the current price of a studio flat is $650$ € per month:

1. a) What will the rental price be in $3$ years' time?
2. b) Identify the elements of the geometric progression modelling this situation (initial capital $C_0$ and common ratio $r$).

A bank offers a compound interest deposit with an annual rate of return of $3 \%$. If we deposit $5000$ € and make no further contributions or withdrawals:

1. a) What will the final capital be after $4$ years?
2. b) How much money will we have earned solely from interest?

The market value of a video game console model depreciates by $12 \%$ with each passing year. If the launch price is $499$ €:

1. a) Write the first three terms of the sequence of annual prices.
2. b) Calculate the value of the console after $5$ years.

Calculate the final capital obtained by investing $1500$ € over $2$ years at annual compound interest of $4 \%$, if the compounding period of interest is monthly (that is, interest is credited each month).

A courier company in Seville purchases an electric delivery van for an initial value of $28\,000$ €. The company knows that this type of vehicle suffers a continuous annual depreciation of $12\,\%$ on its value from the previous year. Marcos, the intern accountant, calculates the value of the vehicle after $4$ years as follows:

"Since the vehicle loses $12\,\%$ each year, in $4$ years it will have lost $4 \cdot 12\,\% = 48\,\%$ of its original value. Therefore, the book value after $4$ years will be $28\,000 \cdot (1 - 0{,}48) = 14\,560$ €".

1. a) Explain in your own words the conceptual error Marcos is making and why his method confuses the linear model (simple interest) with the geometric process of depreciation.
2. b) Model the value of the vehicle as a geometric progression: identify the initial term $a_1$ (value after the first year), the common ratio $r$, and write the general term $a_n$ that determines the value of the vehicle at the end of year $n$.
3. c) Calculate the actual book value of the van after $4$ years and determine by how many euros Marcos overestimated or underestimated the value of the asset.

Sofía deposits $5\,000$ € into a fixed-term deposit account at a Spanish bank that guarantees her an annual compound interest rate of $4\,\%$, without making any further deposits or withdrawals.

1. a) Estimate, before performing any analytical calculation with logarithms, approximately how many years it will take to accumulate at least $7\,000$ € (a $40\,\%$ increase over the initial capital). To do this, make an approximate mental calculation considering only the simple interest of the first year and explain whether the actual time with compound interest will be somewhat longer or shorter than your estimate.
2. b) State the final capital equation $C_n = C_0 \cdot (1+i)^n$ and solve for $n$ using logarithms to calculate the exact number of periods needed to reach or exceed $7\,000$ €.
3. c) Compare the analytical result with your initial estimate and justify what whole number of years Sofía must wait to be able to withdraw at least that amount.

A young woman decides to invest her savings of $2500 €$ in a financial product offering an annual compound interest rate of $3,2 \%$.

1. a) What will the final capital be if interest is compounded annually over 6 years?
2. b) What would the difference be if compounding were monthly instead of annual for that same period of time?

An investor analyses two different savings accounts:

• Account A: Offers an annual interest rate of $4 \%$ with quarterly compounding.

• Account B: Offers an annual interest rate of $4,05 \%$ with annual compounding.

Determine, giving reasons and calculating the Annual Equivalent Rate (AER / TAE) for Account A, which of the two options is more beneficial for the investor.

A small entrepreneur applies for a loan of $15000 €$ to purchase machinery. The bank grants the loan at an annual interest rate of $5,5 \%$, which must be repaid in equal monthly instalments over 4 years.

1. a) Calculate the amount of the monthly instalment to be paid.
2. b) What will the total interest paid amount to by the end of the loan?

A university student buys a latest-generation laptop for $1200\ €$. The value of the computer is estimated to depreciate by $15\,\%$ each year cumulatively.

1. a) What will the value of the computer be after 3 years of use?

1. b) How many years must pass for the value of the computer to be less than a quarter of its original price? (Use logarithms for the calculation).

An 18-year-old wants to build up a fund of $150000 €$ in 20 years' time, when she turns 38, for a house deposit and to start her own business. If the financial institution guarantees her an average annual interest rate of $3,5 \%$ compounded annually:

What fixed amount should she deposit at the beginning of each year to achieve her goal?

Given the geometric progression defined by the terms $a_2 = 12$ and $a_5 = 324$:

1. a) Find the common ratio $r$ and the first term $a_1$.
2. b) Calculate the sum of the first 10 terms of the progression.
3. c) Would it be possible to calculate the sum of its infinite terms? Justify your answer.

A family wants to set up a savings fund for their child's university studies in $4$ years. To do so, they decide to make four constant contributions of $a$ euros at the beginning of each of the next $4$ years into a financial product that yields a guaranteed annual compound interest rate of $3{,}5\,\%$. At the end of the fourth year, the accumulated balance in the fund is exactly $10\,843{,}60$ €.

1. a) Write the equation that models this savings plan relating the final capital to the annual payment $a$, based on the sum of the terms of a geometric progression with common ratio $r = 1 + i$.
2. b) From the stated equation, solve for and calculate the value of the annual contribution $a$.
3. c) State the reverse situation: the family contributes exactly $2\,000$ € at the beginning of each year for $3$ years and wishes to accumulate a final capital of $6\,500$ €. Write the polynomial equation in terms of the ratio $r = 1+i$ that would allow calculating the annual rate of return $i$ required to achieve that goal.

According to data from the National Statistics Institute (INE), a municipality in the Serranía de Cuenca region had $3\,200$ registered inhabitants as of January $1$, $2020$. Due to population ageing and the lack of generational replacement, its population decreases each year at a steady rate of $4\,\%$ relative to the previous year.

1. a) Model the population evolution as a geometric progression: determine the common ratio $r$, write the general term $P_n$ that estimates the population after $n$ years have elapsed since $2020$, and calculate the projected population as of January $1$, $2025$.
2. b) If this trend continues steadily, in which year will the municipality's population drop below $2\,000$ inhabitants for the first time? State the corresponding inequality, solve it by applying logarithms (paying special attention to the properties of inequalities when dividing by negative numbers), and explain your temporal answer.
3. c) Using the formula for the sum of the terms of a geometric progression, calculate how many inhabitants the municipality will have lost in total from January $1$, $2020$ to January $1$, $2030$.

A young entrepreneur decides to open a savings account to finance her sustainable clothing brand project in 5 years' time. To do so, she commits to depositing $150$ € at the beginning of each month. The bank offers her a nominal annual interest rate of $4,5\%$, with monthly compounding of interest.

Calculate the total capital she will have accumulated at the end of the 5 years. To solve this, you must first find the equivalent monthly interest rate and apply the formula for the sum of the terms of a geometric progression (sinking fund annuity).

A family applies for a mortgage loan of $180.000$ € to be repaid over 20 years through constant monthly instalments (French amortisation system). The bank charges annual interest of $3,25\%$. However, the contract includes a "partial interest-only" clause during the first two years, in which the family will only pay the interest accrued, without repaying any principal capital.

1. a) Calculate the monthly instalment that the family will pay during the first two years (grace period).
2. b) Calculate the monthly instalment they must pay from the third year onwards until the end of the loan. Take into account that at that point the outstanding principal is still the initial amount, but the remaining time has decreased.

We compare two investment products for a capital of $10.000$ € over 4 years:

• Product A: Annual compound interest of $5\%$.

• Product B: Nominal annual interest of $4,8\%$ with monthly compounding of interest.

Determine which of the two products is more profitable by calculating the Annual Equivalent Rate (AER / TAE) of each. Explain why, even though Product B has a lower nominal interest rate, the compounding frequency influences the final outcome.

An art collector buys a sculpture for $12.000$ €. It is estimated that its value will increase by $8\%$ each year at a constant rate. At the same time, the average annual inflation forecast for the coming decade is $2,5\%$.

1. a) What will be the nominal value of the sculpture after 10 years?
2. b) What will be the real value of the sculpture after those 10 years, expressed in "today's euros"? To do this, you must deflate the nominal value using the price index derived from inflation (which also follows a geometric progression).

A local council wants to upgrade its street lighting systems to save energy. The cost of the work is $1.500.000$ €. To pay for it, it takes out a loan at $2,8\%$ annual interest, which it will repay through 10 equal annual instalments.

After paying the fifth instalment, the local council receives a European grant and decides to pay off the outstanding debt in a single payment. Calculate how much money the council must pay out at that time to settle the loan completely.

A ball is dropped from an initial height of $10$ metres. On each bounce, the ball reaches a height equal to $2/3$ of the height of the previous bounce.

1. a) What height will it reach after the sixth bounce?
2. b) Calculate the total distance travelled by the ball (upwards and downwards) until it theoretically comes to a stop. Use the sum formula for an infinite decreasing geometric progression, taking into account that, except for the initial drop, all paths are travelled twice (upwards and downwards).

Marta's grandparents want to set up a fund to pay for her university studies. Their goal is for her to receive a monthly income of $500$ € for $4$ years while studying away from home. Assuming that the bank guarantees them annual compound interest of $3\%$ throughout that entire period (the annuity payout period):

1. a) How much capital should they have accumulated when Marta starts university to be able to pay that monthly income? (Hint: Treat this problem as finding the principal of a loan where the monthly instalment is the income she receives).
2. b) If there are still $10$ years before Marta starts university and they currently have no savings, what fixed monthly amount should they deposit into a savings account offering $4\%$ annual interest in order to accumulate the capital calculated in the previous part?

To build up a capital of $25\,000$ € for a down payment on a home in Valencia, two people evaluate two different savings methods at an institution offering an annual compound return of $4\,\%$:

• Strategy A (Lump sum): Deposit a fixed amount of $16\,000$ € today and let it generate interest without making further contributions.

• Strategy B (Savings annuities): Start with no initial capital and contribute a fixed payment of $2\,500$ € at the beginning of each year.

1. a) For Strategy A, state the equation $C_n = C_0 \cdot (1+i)^n$, describe the algebraic method to solve directly for $n$ using logarithms, and calculate the whole number of years needed to exceed the target.
2. b) For Strategy B, formulate the equation from the sum of a geometric progression for annuity-due contributions: $\text{Final capital} = a \cdot (1+i) \cdot \dfrac{(1+i)^n - 1}{i}$. Explain in detail why logarithms cannot be applied immediately in this case, describe what prior algebraic transformations are mandatory to isolate the power $(1+i)^n$, and calculate how many whole years are needed to reach the target.
3. c) Compare both methods and justify which of the two strategies is faster in time and which requires a smaller total out-of-pocket outlay from the saver.

An agricultural cooperative in Extremadura installs a self-consumption photovoltaic plant for a total cost of $120\,000$ €. The degradation of the solar panels gradually reduces the energy produced: the first year it generates a net savings of $15\,000$ €, and in subsequent years the savings generated each year decreases by $5\,\%$ compared to the immediately preceding year. The cooperative deposits this savings in full into a sinking fund with no additional financial return to recover the project cost.

1. a) Model the annual savings $A_n$ corresponding to year $n$ as a geometric progression, indicating the first term $A_1$ and the common ratio $r$.
2. b) Write the expression for the accumulated sum $S_n$ after $n$ years of operation. From it, analyze with justification whether the cooperative will ever fully pay off the installation ($120\,000$ €) if the plant operates indefinitely, studying the behavior of the sum as the number of years $n$ grows indefinitely ($n \to \infty$).
3. c) Suppose the installation company offers a reconditioning warranty that reduces annual degradation to $2\,\%$ (that is, each year it yields $2\,\%$ less than the previous year instead of $5\,\%$), maintaining the initial savings of $15\,000$ € in the first year. Model the new situation and determine, by solving using logarithms in the geometric sum equation, how many years it will take the cooperative to amortize the $120\,000$ € investment.