In The Allowance Choice, the doubling penny looked laughable next to the flat weekly allowance — until it wasn’t, suddenly and by a mile. That collision between steady adding and steady multiplying is the entire mathematics of money, and you already own both halves of it.
Interest is a sequence wearing dollar signs
Simple interest pays the same amount every period: the balance is an arithmetic sequence, and its graph is a straight line. Compound interest pays interest on the interest: each balance is the previous one multiplied by the same factor, which makes the balances a geometric sequence — and the amount function
is The Exponential Function wearing dollar signs. Here is the principal, the interest rate per compounding period, and the number of periods. Watch the gap open — $1000 at 6% for ten years:
| Arrangement | Amount after 10 years |
|---|---|
| Simple interest | $1600.00 |
| Compounded annually | $1790.85 |
| Compounded monthly | $1819.40 |
The compounding column pulls further ahead every year, and compounding more often at the same nominal rate quietly pays more. That is the constant-ratio engine at work.
The formula and its fine print
The fine print is and . “6% per year, compounded monthly for ten years” means and — so . Misreading the period is the classic error here, and it is a productive one: compare the wrong answer with the right one and you feel what compounding frequency does — Mistakes Are Data, with a dollar value attached.
Reverse questions — “how long until it doubles?” — are solved the same honest ways as any exponential question at this stage: systematic guess-and-check on , or trace the graph in Desmos. At 8% compounded annually, lands almost exactly at years.
Annuities — and the other side of the ledger
Almost nobody saves with one deposit. An annuity is a stream of equal deposits at regular intervals — and since each deposit compounds for a different length of time, the future value is a geometric series: the last deposit grows not at all, the first grows longest, and the sum formula adds the whole stream in one line.
Time in the market beats size of deposit, and the gap is not subtle. At 6% compounded annually, $1000 every year from age 20 to 65 becomes about $213,000; $3000 every year from age 50 to 65 — the same $45,000 deposited — becomes about $70,000. The early dollars simply compound longer. A tax-free savings account makes exactly this mathematics available to you within a few years of this course.
Borrowing runs the same formula in reverse — now you are the investment. Leave $2000 unpaid on a credit card at 20% per year, compounded monthly, and two years later the debt is about $2974: nearly a thousand dollars for waiting. When you compare the total interest on any loan with the amount actually borrowed, the formula is neither cruel nor kind — it is just exponential, and it works for whoever owns the principal.
Your Financial Future hands you a spreadsheet and lets you test your own plans against these formulas; Sequences, Series, and Interest Practice builds the fluency first.
Curriculum connection
C3.2
make and describe connections between compound interest, geometric sequences, and exponential growth, through investigation with technology (e.g., use a spreadsheet to make compound interest calculations, determine finite differences in the amounts over time, and graph amount versus time) Sample problem: Describe an investment that could be represented by the function .
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C3.3
solve problems, using a scientific calculator, that involve the calculation of the amount, (also referred to as future value, ), the principal, (also referred to as present value, ), or the interest rate per compounding period, , using the compound interest formula in the form [or ] Sample problem: Two investments are available, one at 6% compounded annually and the other at 6% compounded monthly. Investigate graphically the growth of each investment, and determine the interest earned from depositing $1000 in each investment for 10 years.
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C3.5
explain the meaning of the term annuity, and determine the relationships between ordinary simple annuities (i.e., annuities in which payments are made at the end of each period, and compounding and payment periods are the same), geometric series, and exponential growth, through investigation with technology (e.g., use a spreadsheet to determine and graph the future value of an ordinary simple annuity for varying numbers of compounding periods; investigate how the contributions of each payment to the future value of an ordinary simple annuity are related to the terms of a geometric series) Sample problem: Compare the amounts at age 65 that would result from making an annual deposit of $1000 starting at age 20, or from making an annual deposit of $3000 starting at age 50, to an RRSP that earns 6% interest per annum, compounded annually. What is the total of the deposits in each situation?
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C3.7
solve problems, using technology (e.g., scientific calculator, spreadsheet, graphing calculator), that involve the amount, the present value, and the regular payment of an ordinary simple annuity (e.g., calculate the total interest paid over the life of a loan, using a spreadsheet, and compare the total interest with the original principal of the loan)
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