C. Discrete Functions · Financial Applications: Interest, Present Value and AnnuitiesCompound Interest, Geometric Sequences and Exponential Growth
C3.2About 70 minutes
Learning goals- I can explain how compound interest differs from simple interest.
- I can show that the year-end amounts of a compound-interest investment form a geometric sequence with common ratio 1 + i.
- I can describe compound interest as exponential growth using finite differences, ratios, graphs and the equation A = P(1 + i)ⁿ.
- I can use a spreadsheet to calculate compound interest, find first differences and graph the amount.
Words to know
compound interest
Interest calculated on the principal and on interest already earned, so interest earns interest.
compounding period
The time between the moments when interest is calculated and added (for example one year).
geometric sequence
A sequence in which each term is the previous term multiplied by the same number, the common ratio r.
growth factor
The number you multiply by each period. For compound interest it is 1 + i.
exponential growth
Growth by the same percent (the same factor) in each equal time step. Its graph curves upward and its consecutive values have a constant ratio.
Interest on interest
Go back to your $2 000 birthday gift. This time the bank pays 4% compounded annually. At the end of Year 1 you get $80, just like simple interest. But in Year 2 the bank pays 4% on $2 080, not on $2 000. The $80 you earned last year is now earning interest too.
That small change is the difference between a straight line and a curve that eventually shoots upward. It is why people say "start saving early", and why unpaid credit card balances grow so fast.
Year by year
Each year the amount is multiplied by 1 + i, where i is the rate per compounding period as a decimal. For 4% compounded annually, the growth factor is 1.04.
- Year 1: 2000 × 1.04 = 2080
- Year 2: 2080 × 1.04 = 2163.20
- Year 3: 2163.20 × 1.04 = 2249.728 ≈ 2249.73
In general, after n compounding periods: A = P(1 + i)ⁿ.
Tip: in a calculation, keep the unrounded value and round only the final answer to the cent.
Worked exampleExample 1: Compound interest as a geometric sequence
You invest $2 000 at 4% compounded annually. (a) Show that the year-end amounts form a geometric sequence. (b) Write the general term. (c) Find the amount after 10 years.
- (a) Year-end amounts: $2 080, $2 163.20, $2 249.73, $2 339.72, $2 433.31, … .
- The ratios are 2163.20 ÷ 2080 = 1.04, 2249.728 ÷ 2163.20 = 1.04, and so on. The common ratio is r = 1.04, so the sequence is geometric.
- (b) With the formula from Sequences and Series, tₙ = arⁿ⁻¹ = 2080(1.04)ⁿ⁻¹. Since 2080 = 2000(1.04), this simplifies to tₙ = 2000(1.04)ⁿ.
- (c) t₁₀ = 2000(1.04)¹⁰ ≈ $2 960.49.
Geometric with r = 1.04; tₙ = 2000(1.04)ⁿ; after 10 years about $2 960.49.
Finite differences tell the story
Look at the first differences of the compound amounts (starting at Year 0):
| Year n | Amount | First difference | Ratio to previous amount |
|---|
| 0 | $2 000.00 | | |
| 1 | $2 080.00 | $80.00 | 1.04 |
| 2 | $2 163.20 | $83.20 | 1.04 |
| 3 | $2 249.73 | $86.53 | 1.04 |
| 4 | $2 339.72 | $89.99 | 1.04 |
| 5 | $2 433.31 | $93.59 | 1.04 |
Three facts to notice:
- The first differences are not constant, so compound interest is not linear.
- The second differences (3.20, 3.33, 3.46, 3.60) are not constant either, so it is not quadratic.
- The ratio of consecutive amounts is constant (1.04). A constant ratio means exponential growth.
This is exactly how you told exponential functions apart from linear and quadratic ones in the unit Exponential Functions. In fact, A(n) = 2000(1.04)ⁿ is an exponential function with base 1.04 > 1, so it is increasing, its A-intercept is 2000, and its domain here is the whole numbers n = 0, 1, 2, … (a discrete function).
Worked exampleExample 2: Which kind of interest?
Decide whether each table of year-end amounts shows simple interest, compound interest, or neither. Give the rate when you can.
(a) $400, $420, $441, $463.05
(b) $400, $420, $440, $460
(c) $400, $420, $450, $490
- (a) Differences: 20, 21, 22.05 (not constant). Ratios: 420 ÷ 400 = 1.05, 441 ÷ 420 = 1.05, 463.05 ÷ 441 = 1.05 (constant). Compound interest at 5%.
- (b) Differences: 20, 20, 20 (constant). Simple interest: Pr = 20 and P = 400, so r = 20 ÷ 400 = 5%.
- (c) Differences: 20, 30, 40 (not constant). Ratios: 1.05, about 1.071, about 1.089 (not constant). Neither. (The second differences are constant, 10 and 10, so this pattern is quadratic; it is not how a bank pays interest.)
(a) compound, 5%; (b) simple, 5%; (c) neither.
Worked exampleExample 3: Working backward
An account earning interest compounded annually holds $2 704.00 at the end of Year 2 and $2 812.16 at the end of Year 3. Find the rate and the principal.
- Ratio: 2812.16 ÷ 2704 = 1.04. So 1 + i = 1.04 and the rate is 4%.
- Year 2 amount = P(1.04)², so P = 2704 ÷ 1.04² = 2704 ÷ 1.0816 = $2 500.
4% compounded annually; principal $2 500.
Spreadsheet for compound interest
Change one formula from the simple-interest sheet: in B3 type **=B21.04* (multiply the previous amount). Keep C3 = B3−B2 for first differences and add D3 = B3/B2 for the ratio. Copy down. Column C now grows; column D stays at 1.04. A scatter graph of columns A and B curves upward.
A rule like "each amount is 1.04 times the previous one", with t₀ = 2000, is a recursive formula: t₀ = 2000, tₙ = 1.04tₙ₋₁. The spreadsheet is really building the sequence recursively.
Just for funYou may have read online that Albert Einstein called compound interest "the eighth wonder of the world". Quote researchers have never found any evidence that he said it. The quote has been compounding all by itself.
Hands-on activityRace the two kinds of interest
You need: a spreadsheet app, or a calculator, paper and a pencil.
- In one column build simple interest on $1 000 at 8% for 30 years (add $80 each year). In the next column build compound interest on $1 000 at 8% compounded annually (multiply by 1.08 each year).
- In a third column, find the first differences of the compound column.
- Find the first year when the compound amount is at least double the simple amount. (On a calculator, keep pressing × 1.08 = to step through the compound values.)
- Graph both columns on the same axes.
What to look for: after 30 years, simple interest gives $1 000 + 30 × $80 = $3 400 and compound interest gives 1000 × 1.08³⁰ ≈ $10 062.66, nearly three times as much. The compound amount first reaches double the simple amount in Year 23 (about $5 871.46 compared with $2 840). The compound first differences start at $80 and keep growing.
Your turn
1$1 500 is invested at 5% compounded annually. Find the amounts at the end of years 1 to 4.
$1 575.00, $1 653.75, $1 736.44, $1 823.26 (multiply by 1.05 each year).
2For the investment in the last question, state the common ratio and write the general term tₙ (n = years).
r = 1.05; tₙ = 1500(1.05)ⁿ.
3Is $400, $420, $441, $463.05 simple or compound interest? Explain.
The ratios are all 1.05, so it is compound interest at 5%.
4Is $400, $420, $440, $460 simple or compound interest? Explain.
The first differences are all 20, so it is simple interest (r = 20 ÷ 400 = 5%).
5Find the amount after 10 years for $1 500 at 5% compounded annually.
1500(1.05)¹⁰ ≈ $2 443.34.
6Find the first differences of $1 500, $1 575, $1 653.75, $1 736.44, $1 823.26. What do you notice?
$75.00, $78.75, $82.69, $86.82. They increase, by a factor of about 1.05 each time.
7How much more does compound interest earn than simple interest on $1 500 at 5% over 10 years?
Simple: 1500 + 10 × 75 = $2 250. Compound: $2 443.34. Difference ≈ $193.34.
More practice: warm-up
1In one sentence, what is compound interest?
Interest earned on the principal and on the interest already added, so interest earns interest.
2A growth factor of 1.03 per year means what annual rate?
3$100 at 10% compounded annually. Find the amount after 2 years.
4Find the common ratio of 250, 275, 302.50.
275 ÷ 250 = 1.1 (and 302.50 ÷ 275 = 1.1).
5Which formula gives the amount with compound interest: A = P + Prt or A = P(1 + i)ⁿ?
More practice: on your own
1Year-end amounts are $3 000 (Year 0), $3 120, $3 244.80, $3 374.59. Show that this is compound interest, give the rate, and find the amount after 8 years.
Ratios: 3120 ÷ 3000 = 1.04 and 3244.80 ÷ 3120 = 1.04, so it is compound at 4%. A = 3000(1.04)⁸ ≈ $4 105.71.
2Interest is compounded annually. An account holds $2 704.00 after 2 years and $2 812.16 after 3 years. What will it hold after 4 years?
The ratio is 1.04, so Year 4: 2812.16 × 1.04 ≈ $2 924.65.
3Explain why the first differences of compound-interest amounts get bigger each year.
Each year's interest is i times the current balance, and the balance keeps growing, so each year's interest (the first difference) is larger than the last.
4$5 000 is invested for 20 years at 6%. How much more does compound interest (annually) give than simple interest?
Compound: 5000(1.06)²⁰ ≈ $16 035.68. Simple: 5000 + 5000(0.06)(20) = $11 000. Difference ≈ $5 035.68.
5Write a recursive formula and an explicit formula for $800 invested at 3.5% compounded annually.
Recursive: t₀ = 800, tₙ = 1.035tₙ₋₁. Explicit: tₙ = 800(1.035)ⁿ.
Challenge
1$1 000 is invested at 8% simple interest and another $1 000 at 5% compounded annually. After how many full years is the compound account worth more? Show your check.
Compare 1000(1 + 0.08n) with 1000(1.05)ⁿ. n = 18: $2 440 vs $2 406.62 (simple still ahead). n = 19: $2 520 vs $2 526.95. The compound account is worth more after 19 years, even though its rate is lower.
2Interest is compounded annually. An account holds $2 431.01 after 4 years and $2 680.19 after 6 years. Find the rate and the principal.
Over 2 years the factor is 2680.19 ÷ 2431.01 ≈ 1.1025 = (1 + i)². So 1 + i = √1.1025 = 1.05: the rate is 5%. P = 2431.01 ÷ 1.05⁴ ≈ $2 000.
3Show that the first differences of tₙ = P(1 + i)ⁿ form a geometric sequence with the same ratio, 1 + i.
tₙ₊₁ − tₙ = P(1 + i)ⁿ⁺¹ − P(1 + i)ⁿ = P(1 + i)ⁿ[(1 + i) − 1] = Pi(1 + i)ⁿ. This is a geometric sequence with first term Pi and common ratio 1 + i.
Quiz
1. The year-end amounts of an investment earning compound interest (compounded annually) form:
2. Find the amount after 3 years for $1 000 at 4% compounded annually.
3. What is the common ratio of the year-end amounts for 6% compounded annually?
4. Which list of year-end amounts shows compound interest?
5. Compared with simple interest at the same rate, the graph of compound interest (annual compounding) is:
6. Why does compound interest eventually beat simple interest at the same rate?
Big idea: Expectation C3.2 connects compound interest, geometric sequences and exponential growth, using technology to calculate amounts, find finite differences and graph amount against time.
How to help: Ask, "What do you multiply by each year?" The answer (1 + i) is the key to the whole lesson. A calculator trick helps: type 2000, then × 1.04 = and keep pressing = (on most calculators this repeats the multiplication).
Common difficulties:
- Adding the first year's interest every year (that is simple interest).
- Rounding each year and carrying the rounded value forward; small rounding errors build up. Round only the final answer.
- Confusing the rate i (0.04) with the ratio 1 + i (1.04).
- Deciding between arithmetic and geometric: check differences first, then ratios.