Geometric progressions
Compound interest multiplies the capital by the same factor every year. Saving a fixed amount each month in an account that capitalises also produces a geometric progression, one for each payment. It is the same pattern as in 3rd ESO, now with money involved.
General term and common ratio: compound interest
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…
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.
- a) Determine the common ratio $r$ of the progression.
- 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…
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.
- a) Write the general term $a_n$ of this progression.
- b) Use the sum formula for a geometric progression to find out how much money she will have accumulated in total after $8$ weeks.
The sum of a geometric progression: savings annuities
If you pay a fixed amount $a$ at the start of each period into an account with interest $i$, each payment earns interest for a different number of periods: the last one is capitalised once, the first one is capitalised $n$ times. The total is the sum of a geometric progression.
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…
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).
Finding the number of periods with logarithms
When the unknown is the exponent $n$, you solve for it by taking logarithms on both sides.
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…
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.
- a) Calculate the amount of the monthly instalment to be paid.
- b) What will the total interest paid amount to by the end of the loan?
Problems with financial geometric progressions
Vehicle depreciation, savings plans, comparing bank products: always identify $a_1$, $r=1+i$ and whether you are being asked for a term, a sum or the number of periods.
A young woman decides to invest her savings of $2500 €$ in a financial product offering an annual compound interest rate of $3,2 \%$. a) What will the final capital be…
A young woman decides to invest her savings of $2500 €$ in a financial product offering an annual compound interest rate of $3,2 \%$.
- a) What will the final capital be if interest is compounded annually over 6 years?
- b) What would the difference be if compounding were monthly instead of annual for that same period of time?
A grandfather opens a savings account for his newborn grandson with an initial capital of $2000$ €. The bank offers simple annual interest of $2,5 \%$. a) Calculate the…
A grandfather opens a savings account for his newborn grandson with an initial capital of $2000$ €. The bank offers simple annual interest of $2,5 \%$.
- a) Calculate the interest generated by the account after $10$ years.
- b) What will the total capital accumulated amount to after that period?