Geometric sequences
Maya just started her first part-time job and wants to save for a used car. Her credit union pays compound interest, and she wonders how much a fixed monthly deposit will grow to in 5 years. Compound interest multiplies the balance by the same factor every year, and saving a fixed amount each month also produces a geometric series, one term for each deposit. You may have seen geometric sequences in Algebra 1 as exponential growth; now we use them with money.
General term and common ratio: compound interest
A grandfather opens a savings account for his newborn grandson with an initial deposit of $2000$ dollars. The bank offers simple annual interest of $2.5 \%$. a)…
A grandfather opens a savings account for his newborn grandson with an initial deposit of $2000$ dollars. 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 balance of the account 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 apartment is $650$…
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 $650$ dollars per month:
- a) What will the rental price be in $3$ years' time?
- b) Identify the elements of the geometric sequence modeling this situation (initial amount $C_0$ and common ratio $r$).
A bank offers a compound interest deposit with an annual rate of return of $3 \%$. If we deposit $5000$ dollars and make no further contributions or withdrawals: a) What…
A bank offers a compound interest deposit with an annual rate of return of $3 \%$. If we deposit $5000$ dollars and make no further contributions or withdrawals:
- a) What will the final capital be after $4$ years?
- b) How much money will we have earned solely from interest?
Equivalent rate and APY
Calculate the final capital obtained by investing $1500$ dollars over $2$ years at annual compound interest of $4 \%$, if the compounding period of interest is monthly…
Calculate the final capital obtained by investing $1500$ dollars 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 young woman decides to invest her savings of $\$2500$ in a financial product offering an annual compound interest rate of $3.2 \%…
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?
An investor analyzes two different savings accounts: Account A: Offers an annual interest rate of $4 \%$ with quarterly compounding. Account B: Offers an annual interest…
An investor analyzes two different savings accounts:
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Account A: Offers an annual interest rate of $4 \%$ with quarterly compounding.
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Account B: Offers an annual interest rate of $4.05 \%$ with annual compounding.
Determine, giving reasons and calculating the Annual Percentage Yield (APY) for Account A, which of the two options is more beneficial for the investor.
We compare two investment products for a capital of $10,000$ dollars over 4 years: Product A: Annual compound interest of $5\%$. Product B: Nominal annual interest of…
We compare two investment products for a capital of $10,000$ dollars over 4 years:
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Product A: Annual compound interest of $5\%$.
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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 Percentage Yield (APY) of each. Explain why, even though Product B has a lower nominal interest rate, the compounding frequency influences the final outcome.
The sum of a geometric series: savings annuities
If you deposit a fixed amount $a$ at the start of each period into an account with interest $i$ (think of a monthly deposit into a savings account or a 401(k)), each deposit earns interest for a different number of periods: the last one is compounded once, the first one is compounded $n$ times. The total is the sum of a geometric sequence.
Finding the number of periods
Sometimes the unknown is the exponent $n$: how long until my money reaches a goal? Without logarithms you have two tools. The first is the Rule of 72: money growing at $x\%$ a year doubles in about $72 \div x$ years. The second is a table or a spreadsheet: the NPER function gives the number of periods directly, or you can build a table of balances year by year until you pass the goal.
A college student buys a latest-generation laptop for $\$1200$. The value of the computer is estimated to depreciate by $15\,\%…
A college student buys a latest-generation laptop for $\$1200$. The value of the computer is estimated to depreciate by $15\,\%$ each year cumulatively.
- a) What will the value of the computer be after 3 years of use?
- 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.)
Problems with financial geometric progressions
Car or laptop depreciation, 401(k) and savings plans, CDs, 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.
Back to the start: Maya's balance grows by the same factor $r=1+i$ every period, which is the compound interest pattern from the first section. Her regular monthly deposits form a geometric series: with $\$150$ a month at $0.375\%$ a month, she ends with about $\$10{,}110$ after 5 years, enough for a down payment on a used car.
Real value and nominal value
Prices rise with inflation, so $\$1$ in the future buys less than $\$1$ today. The nominal value is the amount in future dollars. The real value is that amount expressed in today's dollars. Prices, tracked by the Consumer Price Index (CPI), follow a geometric sequence with ratio $1+f$, where $f$ is the inflation rate, so you divide by $(1+f)^n$: $\text{Real value}=\dfrac{\text{Nominal value}}{(1+f)^n}$.
The market value of a video game console model depreciates by $12 \%$ with each passing year. If the launch price is $499$ dollars: a) Write the first three terms of the…
The market value of a video game console model depreciates by $12 \%$ with each passing year. If the launch price is $499$ dollars:
- a) Write the first three terms of the sequence of annual prices.
- b) Calculate the value of the console after $5$ years.
An art collector buys a sculpture for $12,000$ dollars. It is estimated that its value will increase by $8\%$ each year at a constant rate. At the same time, the average…
An art collector buys a sculpture for $12,000$ dollars. 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 $2.5\%$.
- a) What will be the nominal value of the sculpture after 10 years?
- b) What will be the real value of the sculpture after those 10 years, expressed in "today's dollars"? To do this, you must deflate the nominal value using the price index derived from inflation (which also follows a geometric progression).