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.

A grandfather opens a savings account for his newborn grandson with an initial capital of $40\,000$ pesos. 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?

A bank offers a compound interest deposit with an annual rate of return of $3 \%$. If we deposit $5000$ pesos 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?

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

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 apartment is $13\,000$ pesos per month:

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

The market value of a video game console model depreciates by $12 \%$ with each passing year. If the launch price is $9\,980$ pesos:

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

An art collector buys a sculpture for $240\,000$ pesos. 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 pesos"? To do this, you must deflate the nominal value using the price index derived from inflation (which also follows a geometric progression).

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).

A young woman decides to invest her savings of $\$50\,000$ 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.

We compare two investment products for a capital of $200\,000$ pesos 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.

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 $40$ pesos, the second $80$ pesos, the third $160$ pesos, 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.

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 ball is dropped from an initial height of $10$ meters. 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 traveled 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 traveled twice (upwards and downwards).

An 18-year-old wants to build up a fund of $\$3\,000\,000$ 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?

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 $150$ pesos 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 small entrepreneur applies for a loan of $\$300\,000$ 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 family applies for a mortgage loan of $3\,600\,000$ pesos 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.

A municipality wants to upgrade its street lighting systems to save energy. The cost of the work is $30\,000\,000$ pesos. To pay for it, it takes out a loan at $2.8\%$ annual interest, which it will repay through 10 equal annual installments.

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

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 $10\,000$ pesos 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?