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 $2000$ €. 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 small entrepreneur applies for a loan of $\text{€}\ 15000$ 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 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 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.

An 18-year-old wants to build up a fund of $\text{€}\ 150000$ 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?

To build up a capital of $25\,000$ € for a down payment on a home in Galway, 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 $C_n = C_0 \times (1+i)^n$, 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: $\text{Final capital} = a \times (1+i) \times \dfrac{(1+i)^n - 1}{i}$. 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.

Aoife deposits $5\,000$ € into a fixed-term deposit account at an Irish 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 $C_n = C_0 \times (1+i)^n$ 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 Aoife must wait to be able to withdraw at least that amount.

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 $10$ 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).

A young woman decides to invest her savings of $\text{€}\ 2500$ 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.

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.