Arithmetic and geometric progressions

Overview

A progression is a sequence whose terms follow a steady rule. In an arithmetic progression, the same number is added each time. In a geometric progression, each term is multiplied by the same number each time. These two ideas help learners describe number patterns, find missing terms, and calculate sums without listing every term one by one.

This chapter focuses on recognizing the pattern, choosing the correct formula, and explaining why a sequence is or is not an arithmetic progression or a geometric progression.

Prerequisites

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

• Substitution into algebraic expressions.
• Solving simple linear equations.
• Powers and indices, especially repeated multiplication.
• Expanding brackets and simplifying expressions.
• Using sigma-free sum notation such as "sum of the first ten terms."

Learning Scope

This page covers arithmetic progressions, geometric progressions, nth terms, missing terms, common differences, common ratios, and sums of finite progressions. It also covers how to test whether a given sequence belongs to either type.

This page does not treat arithmetic mean, geometric mean, or compound interest in detail; those are handled on Arithmetic mean, geometric mean, and compound interest. Infinite geometric series and advanced summation methods are beyond this Form III page.

Subtopics

Sequences And Terms

A sequence is an ordered list of numbers. Each number in the list is a term. If the position of a term is $n$, then the term in that position is often written as $T_n$.

For example, in the sequence $4, 7, 10, 13, \ldots$, the first term is $T_1 = 4$ and the fourth term is $T_4 = 13$.

Key insight: always connect a term to its position. Many errors happen when learners treat "fifth term" as the number 5 instead of the term at position 5. A helpful reading habit is to say, "position first, value second." In $T_4 = 13$, the position is $4$ and the value is $13$.

When a question gives a list, first write what is known. For $4, 7, 10, 13, \ldots$, write $T_1 = 4$, $T_2 = 7$, $T_3 = 10$, and $T_4 = 13$. This slows the work enough to prevent choosing a formula before the pattern has been tested.

Arithmetic Progressions

An arithmetic progression, or A.P., is a sequence with a constant difference between consecutive terms. If the first term is $a$ and the common difference is $d$, then:

$$ T_n = a + (n - 1)d $$

The sum of the first $n$ terms is:

$$ S_n = \frac{n}{2}\left[2a + (n - 1)d\right] $$

or, if the last term $l$ is known:

$$ S_n = \frac{n}{2}(a + l) $$

Example pattern: $6, 10, 14, 18, \ldots$ is arithmetic because each term increases by $4$.

Why the nth-term formula works: the first term $a$ is already present at position $1$. To reach position $n$, the common difference has been added only $n - 1$ times. That is why the formula uses $(n - 1)d$, not $nd$.

Why the sum formula works: the first and last terms of an A.P. pair neatly. In $4, 9, 14, 19, 24$, the pairs $4 + 24$ and $9 + 19$ have the same total. The formula uses this equal-pair idea to avoid adding every term separately.

Geometric Progressions

A geometric progression, or G.P., is a sequence with a constant ratio between consecutive terms. If the first term is $a$ and the common ratio is $r$, then:

$$ T_n = ar^{n - 1} $$

For $r \ne 1$, the sum of the first $n$ terms is:

$$ S_n = \frac{a(r^n - 1)}{r - 1} $$

Equivalently:

$$ S_n = \frac{a(1 - r^n)}{1 - r} $$

Example pattern: $3, 6, 12, 24, \ldots$ is geometric because each term is multiplied by $2$.

Why the nth-term formula works: the first term $a$ is at position $1$. To reach $T_2$, multiply once by $r$; to reach $T_3$, multiply twice by $r$; to reach $T_n$, multiply $n - 1$ times. That is why the power is $n - 1$.

Why the sum formula works: if $S_n = a + ar + ar^2 + \cdots + ar^{n - 1}$, then multiplying by $r$ shifts every term one place. Subtracting the two sums cancels the middle terms and leaves only the first and final shifted term. This cancellation is the reason the compact formula is possible.

Testing A Sequence

To test for an A.P., subtract consecutive terms:

$$ T_2 - T_1,\quad T_3 - T_2,\quad T_4 - T_3 $$

If the differences are equal, the sequence is arithmetic.

To test for a G.P., divide consecutive terms, where division is defined:

$$ \frac{T_2}{T_1},\quad \frac{T_3}{T_2},\quad \frac{T_4}{T_3} $$

If the ratios are equal, the sequence is geometric.

Key insight: a sequence can fail both tests. Not every sequence with a formula is an A.P. or G.P.

Use enough terms to be confident. Equal first two differences or ratios alone are not enough if more terms are available. Check at least three consecutive differences or ratios when the question gives four or more terms.

Finding Unknown Terms

When terms are given at different positions, use the nth-term formula to form equations. For an A.P., two known terms can determine $a$ and $d$. For a G.P., two known non-zero terms can determine $a$ and $r$.

If three consecutive terms of a G.P. are given as $p$, $q$, and $s$, then:

$$ q^2 = ps $$

This follows because $\frac{q}{p} = \frac{s}{q}$.

For unknown-term questions, write one equation for each known position. Then solve the equations before finding the requested value. A good checking routine is to substitute the values of $a$, $d$, or $r$ back into the original known terms, not only into the final answer.

Key Terms

• Sequence: an ordered list of terms.
• Term: one number or expression in a sequence.
• Progression: a sequence that follows a regular rule.
• Arithmetic progression: a progression with constant difference.
• Common difference: the fixed difference $d$ in an A.P.
• Geometric progression: a progression with constant ratio.
• Common ratio: the fixed multiplier $r$ in a G.P.
• First term: the value $a$ when $n = 1$.
• nth term: the general expression for the term in position $n$.
• Finite sum: the total of a fixed number of terms.

Worked Examples

Example 1: Find A Term In An A.P.

The first term of an A.P. is $5$ and the common difference is $3$. Find the tenth term.

Use $T_n = a + (n - 1)d$.

$$ \begin{aligned} T_{10} &= 5 + (10 - 1)3 \\ &= 5 + 27 \\ &= 32 \end{aligned} $$

The tenth term is $32$.

Check: listing by repeated addition gives $5, 8, 11, 14, 17, 20, 23, 26, 29, 32$. The tenth listed term agrees with the formula.

Example 2: Find The Sum Of An A.P.

Find the sum of the first $12$ terms of the A.P. $4, 9, 14, 19, \ldots$.

Here $a = 4$, $d = 5$, and $n = 12$.

$$ \begin{aligned} S_{12} &= \frac{12}{2}\left[2(4) + (12 - 1)5\right] \\ &= 6(8 + 55) \\ &= 6 \times 63 \\ &= 378 \end{aligned} $$

The sum of the first $12$ terms is $378$.

Check: the twelfth term is $T_{12} = 4 + 11(5) = 59$. The alternative formula gives $S_{12} = \frac{12}{2}(4 + 59) = 6 \times 63 = 378$.

Example 3: Find A G.P. From Two Terms

The fifth and sixth terms of a G.P. are $162$ and $486$. Find the common ratio and first term.

Since consecutive terms are given, divide the sixth term by the fifth term:

$$ r = \frac{486}{162} = 3 $$

Now use $T_5 = ar^4$.

$$ \begin{aligned} 162 &= a(3^4) \\ 162 &= 81a \\ a &= 2 \end{aligned} $$

The common ratio is $3$ and the first term is $2$.

Check: $T_5 = 2(3^4) = 162$ and $T_6 = 2(3^5) = 486$, so both given terms are recovered.

Example 4: Decide Whether A Sequence Is A.P. Or G.P.

The general term of a sequence is $T_n = n(2n - 1)$. Write the first four terms and decide whether it is an A.P. or G.P.

$$ \begin{aligned} T_1 &= 1(2 \times 1 - 1) = 1 \\ T_2 &= 2(2 \times 2 - 1) = 6 \\ T_3 &= 3(2 \times 3 - 1) = 15 \\ T_4 &= 4(2 \times 4 - 1) = 28 \end{aligned} $$

The first four terms are $1, 6, 15, 28$.

Differences:

$$ 6 - 1 = 5,\quad 15 - 6 = 9,\quad 28 - 15 = 13 $$

Ratios:

$$ \frac{6}{1} = 6,\quad \frac{15}{6} = \frac{5}{2},\quad \frac{28}{15} $$

The differences are not equal and the ratios are not equal. The sequence is neither an A.P. nor a G.P.

Example 5: Find An A.P. From Two Non-Consecutive Terms

The fourth term of an A.P. is $13$ and the ninth term is $28$. Find the first term and common difference.

Use $T_n = a + (n - 1)d$.

For the fourth term:

$$ a + 3d = 13 $$

For the ninth term:

$$ a + 8d = 28 $$

Subtract the first equation from the second:

$$ \begin{aligned} (a + 8d) - (a + 3d) &= 28 - 13 \\ 5d &= 15 \\ d &= 3 \end{aligned} $$

Substitute $d = 3$ into $a + 3d = 13$:

$$ \begin{aligned} a + 3(3) &= 13 \\ a + 9 &= 13 \\ a &= 4 \end{aligned} $$

The first term is $4$ and the common difference is $3$.

Check: $T_4 = 4 + 3(3) = 13$ and $T_9 = 4 + 8(3) = 28$.

Example 6: Find Possible Ratios In A G.P.

The third term of a G.P. is $12$ and the fifth term is $48$. Find the possible common ratios.

Use $T_n = ar^{n - 1}$.

$$ T_3 = ar^2 = 12 $$

$$ T_5 = ar^4 = 48 $$

Divide the fifth-term equation by the third-term equation:

$$ \frac{ar^4}{ar^2} = \frac{48}{12} $$

So:

$$ r^2 = 4 $$

Therefore:

$$ r = 2 \quad \text{or} \quad r = -2 $$

If $r = 2$, then $ar^2 = 12$ gives $4a = 12$, so $a = 3$.

If $r = -2$, then $ar^2 = 12$ also gives $4a = 12$, so $a = 3$.

The possible common ratios are $2$ and $-2$.

Check: for $a = 3$ and $r = 2$, the terms include $12$ as $T_3$ and $48$ as $T_5$. For $a = 3$ and $r = -2$, the terms are $3, -6, 12, -24, 48$, so the given terms still match.

Example 7: Use The G.P. Middle-Term Property

If $4$, $x$, and $36$ are consecutive terms of a G.P., find the possible values of $x$.

For three consecutive G.P. terms $p$, $q$, and $s$:

$$ q^2 = ps $$

Here $p = 4$, $q = x$, and $s = 36$.

$$ \begin{aligned} x^2 &= 4 \times 36 \\ x^2 &= 144 \\ x &= 12 \quad \text{or} \quad x = -12 \end{aligned} $$

The possible values are $12$ and $-12$.

Check: $4, 12, 36$ has common ratio $3$. Also $4, -12, 36$ has common ratio $-3$ because $-12 \div 4 = -3$ and $36 \div -12 = -3$.

Common Mistakes

• Using the A.P. formula for a G.P. because both have "first term" and "nth term." Correction: test differences and ratios before choosing a formula. Warning sign: the sequence changes by multiplication, but the working uses $+d$.
• Forgetting that $T_n = a + (n - 1)d$, not $a + nd$. Correction: remember that $a$ is already $T_1$, so only $n - 1$ differences are added. Warning sign: your second term becomes $a + 2d$ instead of $a + d$.
• Treating the common ratio as a difference. Correction: for a G.P., divide consecutive terms. Warning sign: the word "ratio" appears, but the working contains subtraction.
• Dividing by zero when testing ratios. Correction: if a term used as a divisor is zero, the ratio test must be handled carefully; do not write a fraction with zero denominator.
• Assuming every sequence with a visible pattern must be either arithmetic or geometric. Correction: a pattern such as square numbers has a rule but is neither A.P. nor G.P. Warning sign: differences and ratios are both changing.
• Using the sum formula when the question asks for a single term. Correction: underline whether the question asks for "$T_n$" or "sum." Warning sign: your answer is much larger than the terms in the sequence.
• Losing negative signs when the common difference or common ratio is negative. Correction: keep brackets around negative values, especially in powers such as $(-2)^4$.
• Counting $n$ incorrectly when terms are inserted or when a later term is named. Correction: count positions aloud or make a small position table. Warning sign: the first term is counted as term $0$ without the formula being adjusted.
• Rounding too early in G.P. problems. Correction: keep fractions or exact roots until the final step when possible.

Practice Tasks

Foundation

1. State whether each sequence is an A.P., a G.P., or neither: $2, 5, 8, 11, \ldots$; $3, 9, 27, 81, \ldots$; $1, 4, 9, 16, \ldots$.
2. For $6, 11, 16, 21, \ldots$, write $a$, $d$, $T_1$, $T_2$, and $T_4$.
3. For $5, 10, 20, 40, \ldots$, write $a$, $r$, $T_1$, $T_2$, and $T_4$.
4. Find the next three terms of the A.P. $18, 15, 12, \ldots$.

Skill-Building

1. Find the twentieth term of the A.P. whose first term is $7$ and common difference is $-2$.
2. Find the sum of the first $15$ terms of $3, 8, 13, 18, \ldots$.
3. Find the eighth term of the G.P. $5, 10, 20, 40, \ldots$.
4. Find the sum of the first $6$ terms of the G.P. $2, 6, 18, \ldots$.
5. The fourth term of an A.P. is $13$ and the ninth term is $28$. Find the first term and common difference.
6. The third term of a G.P. is $12$ and the fifth term is $48$. Find the possible common ratios.

Exam-Style

1. If $4$, $x$, and $36$ are consecutive terms of a G.P., find the possible values of $x$.
2. A learner says $2, 6, 12, 20, \ldots$ is an A.P. because it is increasing. Explain the error and show the correct test.
3. A sequence has general term $T_n = 3n^2 - 2$. Write the first four terms and decide whether it is an A.P., a G.P., or neither.
4. The sum of the first $n$ terms of an A.P. is $S_n = 2n^2 + 3n$. Find the first three terms, then find the common difference.

Challenge

1. Create an A.P. with tenth term $50$ and common difference $4$. Show how you checked it.
2. Create a G.P. with fourth term $81$ and common ratio $3$. Show how you checked it.
3. The second, fourth, and eighth terms of an A.P. are added. Express the result in terms of $a$ and $d$, then simplify.
4. A G.P. has positive terms, $T_2 = 6$, and $T_5 = 162$. Find $a$ and $r$.

Generated Question Layer

Original generated practice for this topic should include:

• Recognition questions: classify sequences as A.P., G.P., or neither.
• Direct formula questions: find $T_n$ or $S_n$ from $a$, $d$, $r$, and $n$.
• Reverse questions: find $a$, $d$, or $r$ from two known terms.
• Consecutive-term questions: use the middle-term property in a G.P.
• Explanation questions: justify why a sequence does or does not fit a progression.
• Mixed-topic questions: combine progressions with ratios, indices, or simple equations.

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

Learner Aid Opportunities

• planning-marker:chart: comparison table for A.P. versus G.P. definitions, tests, nth-term formulas, sum formulas, and warning signs.
• planning-marker:interactive: term generator where learners adjust $a$, $d$, $r$, and $n$, then see a position table and a formula substitution line.
• planning-marker:LLM tutor: guided diagnosis that asks learners to compute differences first, ratios second, then choose or reject a progression type.
• planning-marker:animation: step-by-step build of terms showing repeated addition for A.P. and repeated multiplication for G.P.
• planning-marker:checklist: printable solution routine for "identify, choose formula, substitute, solve, check against given terms."

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.

| Year | Question ID | Current Mapping Signal | Review Status | | ---: | --- | --- | --- | | 2021 | csee_041_2021_p1_q08_b | G.P. terms used to find the common ratio and first term. | mapped_unreviewed; multi-topic review flag. | | 2022 | csee_041_2022_p1_q08_a | A.P. term information used to find a finite sum. | mapped_unreviewed. | | 2023 | csee_041_2023_p1_q08_a | First terms from a general term and classification as A.P. or G.P. | mapped_unreviewed. | | 2023 | csee_041_2023_p1_q08_b | A.P. sum and selected terms. | mapped_unreviewed. | | 2024 | csee_041_2024_p1_q08_a | Repeated classification task from a general term. | mapped_unreviewed. | | 2025 | csee_041_2025_p1_q08_b | Consecutive G.P. terms with an unknown middle expression. | mapped_unreviewed; missing-marks review flag. |

| Source | Current Signal | Review Status | Use Carefully As | | --- | --- | --- | --- | | data/topic_frequency_2021_2025.json | This topic appears in 6 extracted question records across 2021-2025. | Unreviewed aggregate. | Rough retrieval priority, not a final frequency claim. | | data/exam_format_topic_crosswalk_2022.jsonl | The 2022 examination-format crosswalk maps Sequences and series to this topic as an official format-topic grouping. | Official format mapping; topic-page use still unreviewed. | Evidence that progressions belong to a tested format group. |

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.
• Exam-derived mappings are assessment signals only and remain unreviewed unless a maintainer verifies them against source papers and marking schemes.
• Formula coverage should be checked against the exact Form III teaching sequence used by the school or syllabus guide.