Arithmetic mean, geometric mean, and compound interest

Overview

Arithmetic mean, geometric mean, and compound interest all describe a kind of "middle" or repeated growth. The arithmetic mean balances numbers by addition. The geometric mean balances positive numbers by multiplication. Compound interest uses repeated percentage growth, so it naturally connects to geometric progressions.

This chapter helps learners choose the correct mean, interpret growth over several periods, and solve financial problems where the amount changes by the same percentage each period.

Prerequisites

Before studying this chapter, a learner should be comfortable with:

• Arithmetic progressions and geometric progressions.
• Percentages and percentage increase.
• Powers and roots.
• Solving simple equations.
• Substitution into formulas.
• Basic financial words such as principal, amount, rate, and time.

Learning Scope

This page covers arithmetic mean, geometric mean, inserting means between two numbers, and compound interest for fixed percentage growth over equal time periods.

It does not cover full banking products, depreciation schedules, inflation modelling, annuities, or advanced finance. It also does not replace the full treatment of progressions on Arithmetic and geometric progressions.

Subtopics

Arithmetic Mean

The arithmetic mean of two numbers is the number halfway between them by addition. For two numbers $a$ and $b$, the arithmetic mean is:

$$ \frac{a + b}{2} $$

For several numbers, add all the numbers and divide by how many numbers there are.

Key insight: the arithmetic mean is linked to equal differences. If $a$, $m$, and $b$ form an A.P., then $m$ is the arithmetic mean of $a$ and $b$.

Why this works: if $m$ is exactly halfway between $a$ and $b$, then the distance from $a$ to $m$ equals the distance from $m$ to $b$. That means $m - a = b - m$. Rearranging gives $2m = a + b$, so $m = \frac{a + b}{2}$.

For several values, the arithmetic mean is a fair-share value. Imagine combining all the values into one total and then sharing that total equally among the values. This is why the method is "add, then divide by how many."

Geometric Mean

The geometric mean of two positive numbers $a$ and $b$ is the positive number $g$ such that $a$, $g$, and $b$ form a G.P. Therefore:

$$ g^2 = ab $$

So:

$$ g = \sqrt{ab} $$

Key insight: the geometric mean is linked to equal ratios. It is most useful when quantities grow by multiplication, not by addition.

Why this works: if $a$, $g$, and $b$ form a G.P., then the ratio from $a$ to $g$ equals the ratio from $g$ to $b$. So $\frac{g}{a} = \frac{b}{g}$. Cross-multiplying gives $g^2 = ab$. For positive numbers, the positive middle value is $g = \sqrt{ab}$.

The geometric mean should be used only when the context makes multiplication meaningful, such as repeated growth, scale factors, ratios, or compound interest. For ordinary additive totals, the arithmetic mean is usually the intended average.

Inserting Means

To insert arithmetic means between two numbers, make an A.P. from the first number to the last number. If there are $k$ means between two endpoints, then there are $k + 2$ terms altogether.

To insert geometric means between two positive numbers, make a G.P. from the first number to the last number. Use the nth-term formula to find the common ratio.

Counting is the main bridge. The inserted means are not the only terms; the two endpoints are also terms. If three means are inserted, there are five terms altogether. If two means are inserted, there are four terms altogether. This count determines the position number used in the nth-term formula.

Compound Interest

Compound interest means interest is added to the principal, then the next period's interest is calculated on the new amount. If the principal is $P$, rate is $r\%$ per period, and time is $n$ periods, then the amount is:

$$ A = P\left(1 + \frac{r}{100}\right)^n $$

The compound interest earned is:

$$ I = A - P $$

Key insight: compound interest is a geometric progression because the amount is repeatedly multiplied by the same factor.

Why this works: after one period, the amount is $P\left(1 + \frac{r}{100}\right)$. After two periods, the new amount is multiplied by the same factor again. Repeating the same multiplier creates the powers in the formula.

The rate and the time must use the same period. If the rate is per year, then $n$ must count years. If the rate is per month, then $n$ must count months.

Comparing Simple Growth And Compound Growth

In simple interest, interest is calculated from the original principal each time. In compound interest, interest is calculated from the latest amount. Compound growth is usually larger when the rate is positive and time is more than one period.

For compound interest, the multiplier is:

$$ 1 + \frac{r}{100} $$

For example, a rate of $8\%$ gives multiplier $1.08$.

A quick comparison helps: simple interest repeats the same added amount, so it behaves like an A.P. in the total amount. Compound interest repeats the same multiplier, so it behaves like a G.P. in the total amount.

Key Terms

• Arithmetic mean: the additive average, often found by dividing a sum by the number of values.
• Geometric mean: the multiplicative average, found from equal ratios.
• Principal: the original amount of money invested or borrowed.
• Rate: the percentage used to calculate interest for each period.
• Period: one equal time interval, such as one year.
• Amount: the final value after interest has been added.
• Compound interest: interest calculated on the current amount, including earlier interest.
• Growth factor: the multiplier $1 + \frac{r}{100}$ used in compound interest.

Worked Examples

Example 1: Find An Arithmetic Mean

Find the arithmetic mean of $14$ and $38$.

$$ \begin{aligned} \text{Arithmetic mean} &= \frac{14 + 38}{2} \\ &= \frac{52}{2} \\ &= 26 \end{aligned} $$

The arithmetic mean is $26$. The sequence $14, 26, 38$ has equal differences of $12$.

Check: $26 - 14 = 12$ and $38 - 26 = 12$, so $26$ is halfway by addition.

Example 2: Insert Arithmetic Means

Insert three arithmetic means between $5$ and $25$.

There will be $5$ terms altogether:

$$ 5,\ \_,\ \_,\ \_,\ 25 $$

Use $T_5 = a + 4d$.

$$ \begin{aligned} 25 &= 5 + 4d \\ 20 &= 4d \\ d &= 5 \end{aligned} $$

The sequence is:

$$ 5,\ 10,\ 15,\ 20,\ 25 $$

The three arithmetic means are $10$, $15$, and $20$.

Check: the full sequence has equal differences: $10 - 5 = 5$, $15 - 10 = 5$, $20 - 15 = 5$, and $25 - 20 = 5$.

Example 3: Find A Geometric Mean

Find the positive geometric mean of $4$ and $100$.

$$ \begin{aligned} g &= \sqrt{4 \times 100} \\ &= \sqrt{400} \\ &= 20 \end{aligned} $$

The positive geometric mean is $20$. The sequence $4, 20, 100$ has common ratio $5$.

Check: $20 \div 4 = 5$ and $100 \div 20 = 5$, so the ratios are equal.

Example 4: Insert Geometric Means

Insert two positive geometric means between $3$ and $81$.

There will be $4$ terms altogether:

$$ 3,\ \_,\ \_,\ 81 $$

Use $T_4 = ar^3$.

$$ \begin{aligned} 81 &= 3r^3 \\ 27 &= r^3 \\ r &= 3 \end{aligned} $$

The sequence is:

$$ 3,\ 9,\ 27,\ 81 $$

The two geometric means are $9$ and $27$.

Check: $9 \div 3 = 3$, $27 \div 9 = 3$, and $81 \div 27 = 3$.

Example 5: Calculate Compound Interest

A principal of Tsh $200,000$ is invested at $10\%$ compound interest per year for $3$ years. Find the amount and the compound interest.

$$ \begin{aligned} A &= P\left(1 + \frac{r}{100}\right)^n \\ &= 200000\left(1 + \frac{10}{100}\right)^3 \\ &= 200000(1.1)^3 \\ &= 200000 \times 1.331 \\ &= 266200 \end{aligned} $$

The amount is Tsh $266,200$.

$$ \begin{aligned} I &= A - P \\ &= 266200 - 200000 \\ &= 66200 \end{aligned} $$

The compound interest is Tsh $66,200$.

Check by period:

$$ \begin{aligned} \text{After 1 year} &= 200000 \times 1.1 = 220000 \\ \text{After 2 years} &= 220000 \times 1.1 = 242000 \\ \text{After 3 years} &= 242000 \times 1.1 = 266200 \end{aligned} $$

The period-by-period amount agrees with the formula.

Example 6: Find A Principal From Compound Interest

An amount of Tsh $242,000$ is obtained after investing money at $10\%$ compound interest per year for $2$ years. Find the principal.

Use:

$$ A = P\left(1 + \frac{r}{100}\right)^n $$

Here $A = 242000$, $r = 10$, and $n = 2$.

$$ \begin{aligned} 242000 &= P\left(1 + \frac{10}{100}\right)^2 \\ 242000 &= P(1.1)^2 \\ 242000 &= 1.21P \\ P &= \frac{242000}{1.21} \\ P &= 200000 \end{aligned} $$

The principal was Tsh $200,000$.

Check: Tsh $200,000$ compounded at $10\%$ for two years becomes $200000 \times 1.21 = 242000$.

Example 7: Compare Arithmetic Mean And Geometric Mean

Compare the arithmetic mean and positive geometric mean of $4$ and $36$.

Arithmetic mean:

$$ \begin{aligned} \frac{4 + 36}{2} &= \frac{40}{2} \\ &= 20 \end{aligned} $$

Geometric mean:

$$ \begin{aligned} \sqrt{4 \times 36} &= \sqrt{144} \\ &= 12 \end{aligned} $$

The arithmetic mean is $20$ and the geometric mean is $12$, so the arithmetic mean is larger.

Check: $4, 20, 36$ has equal additive distances of $16$, while $4, 12, 36$ has equal ratios of $3$.

Example 8: Choose The Correct Mean From Context

A machine output is multiplied by the same scale factor each day. It changes from $8$ units on Monday to $72$ units on Wednesday, with Tuesday between them. Estimate Tuesday's output if the growth is multiplicative.

Because the change is multiplicative, use the geometric mean:

$$ \begin{aligned} g &= \sqrt{8 \times 72} \\ &= \sqrt{576} \\ &= 24 \end{aligned} $$

Tuesday's output is $24$ units.

Check: $24 \div 8 = 3$ and $72 \div 24 = 3$, so the same multiplier was used for each step.

Common Mistakes

• Using arithmetic mean when the situation is about repeated multiplication. Correction: use geometric mean for equal ratios, scale factors, and compound growth. Warning sign: the context says "multiplied by" or "same percentage."
• Forgetting to divide by the number of values when finding an arithmetic mean. Correction: after adding, divide by the count of values. Warning sign: the answer is larger than all the given values when an ordinary mean should be between them.
• Writing $\sqrt{a + b}$ instead of $\sqrt{ab}$ for the geometric mean. Correction: geometric mean uses the product, not the sum. Warning sign: the working says "geometric" but uses addition inside the square root.
• Using $r$ as a whole number in the compound interest formula without dividing by $100$. Correction: $8\%$ becomes $0.08$, so the growth factor is $1.08$. Warning sign: using $(1 + 8)^n$ for $8\%$.
• Calculating interest only once when the question asks for compound interest over several periods. Correction: multiply by the growth factor once for each period. Warning sign: the interest added each year is unchanged even though the amount has increased.
• Confusing amount $A$ with interest $I$. Correction: amount is final total; compound interest is $A - P$. Warning sign: the final answer is labelled "interest" but still includes the original principal.
• Treating the number of inserted means as the number of terms instead of counting the two endpoints as well. Correction: $k$ inserted means gives $k + 2$ total terms.
• Mismatching rate period and time period. Correction: if the rate is per year, count years; if the rate is per month, count months. Warning sign: using $n = 2$ for two years when the rate was monthly.
• Rounding the amount before subtracting the principal. Correction: keep exact or enough decimal places until the final currency answer.

Practice Tasks

Foundation

1. Find the arithmetic mean of $18$ and $42$.
2. Find the arithmetic mean of $6$, $10$, $14$, and $18$.
3. Find the positive geometric mean of $9$ and $64$.
4. A value grows by $20\%$ each year. Write its yearly growth factor.

Skill-Building

1. Insert four arithmetic means between $2$ and $27$.
2. Insert two positive geometric means between $2$ and $54$.
3. Find the amount when Tsh $150,000$ is invested at $8\%$ compound interest per year for $2$ years.
4. A principal of Tsh $500,000$ becomes Tsh $605,000$ after compound interest. Find the interest earned.
5. Find the principal that becomes Tsh $363,000$ after $2$ years at $10\%$ compound interest per year.

Exam-Style

1. Explain why compound interest amounts form a G.P. when the rate is fixed.
2. A learner finds the geometric mean of $16$ and $81$ by calculating $\frac{16 + 81}{2}$. Explain the error and find the correct positive geometric mean.
3. Compare the arithmetic mean and geometric mean of $4$ and $36$. Which is larger?
4. Insert three arithmetic means between $12$ and $40$, then check the common difference.
5. Insert one positive geometric mean between $5$ and $80$, then check the common ratio.

Challenge

1. A sum of money is invested at $5\%$ compound interest per year and becomes Tsh $441,000$ after $2$ years. Find the principal.
2. A quantity changes from $12$ to $108$ in two equal multiplicative steps. Find the middle value and the common ratio.
3. Give one situation where the arithmetic mean is suitable and one situation where the geometric mean is suitable. Explain the difference.
4. A learner says compound interest is an A.P. because the amount increases every year. Use a short example to correct the statement.

Generated Question Layer

Original generated practice for this topic should include:

• Direct mean questions: compute arithmetic and geometric means.
• Insertion questions: place one or more means between two endpoints.
• Interpretation questions: choose arithmetic mean or geometric mean from a short context.
• Compound interest formula questions: find amount, interest, principal, rate, or time when enough information is given.
• Error-analysis questions: distinguish $A$, $P$, $r$, $n$, and $I$.
• Link questions: explain how compound interest forms a geometric progression.

Generated questions should be clearly separated from official past-paper questions.

Learner Aid Opportunities

• planning-marker:chart: comparison organizer for arithmetic mean, geometric mean, simple interest, and compound interest with formula, clue words, and checking method.
• planning-marker:interactive: compound interest calculator with adjustable principal, rate, and periods plus a period-by-period table.
• planning-marker:animation: amount growing period by period to show the same multiplier being applied repeatedly.
• planning-marker:LLM tutor: guided hints for deciding whether a problem is additive or multiplicative before selecting a mean or interest formula.
• planning-marker:checklist: solution routine for "identify values, count terms or periods, choose formula, substitute, calculate, label amount or interest."

These are planning markers only; no learner-aid media is currently attached to this page.

Exam-Derived Signals

These signals are assessment leads, not verified official past-question links. They should be checked against original papers and marking schemes before being used as final learner-facing references.

| Source | Current Signal | Review Status | Use Carefully As | | --- | --- | --- | --- | | data/topic_frequency_2021_2025.json | This topic is listed among low-or-no-coverage topics in the extracted 2021-2025 Basic Mathematics question records. | Unreviewed aggregate. | A signal that direct extracted coverage may be sparse. | | data/question_map_2021_2025.jsonl | No direct topic-arithmetic-mean-geometric-mean-and-compound-interest entries were found during this expansion pass. | No reviewed links. | Do not infer absence from the official syllabus; only the local mapping layer lacks direct reviewed links. | | data/curriculum_map.json | The topic is officially present in the 2023 curriculum map. | Official syllabus-derived topic record. | Curriculum coverage evidence, not exam-frequency evidence. |

Source And Review Notes

• Official syllabus and curriculum-map metadata are preserved from data/curriculum_map.json.
• Learner explanations, worked examples, and practice tasks are original prose and should receive subject-teacher review.
• Compound-interest examples use ordinary fixed-period compounding only; any local convention about currency formatting, rounding, or time periods should be reviewed.
• Exam-derived notes are assessment signals only and remain unreviewed.