ECZ GRADE 12 & GCE MATHEMATICS

How to Solve Arithmetic & Geometric Progressions (AP & GP)

Master Section A of Paper 2: finding the $n^{\text{th}}$ term, common difference, common ratio, sum of $n$ terms, and sum to infinity with full working from past ECZ exam papers.

14 Min Read Paper 2 Section A 6 Marks Guaranteed
Exam Candidate Strategy

In ECZ Mathematics Paper 2 (Syllabus 4024), Question 2(b) or Question 3(a) is virtually always a progression question carrying 5 or 6 marks. These are routine calculation marks. By memorizing the 5 essential formulas and showing every step of substitution, you can easily secure 100% of these marks.

Teaching Note

Remind candidates that the formula page on the front of Paper 2 provides formulas for $S_n$ and $S_\infty$, but candidates must correctly distinguish whether a given numerical sequence has a common difference ($d$) or a common ratio ($r$).

Table of Contents

1. Master Formula Sheet: AP vs GP

Before solving any progression question, write down your known values ($a, d, r, n$). Here is your complete reference toolkit:

Arithmetic Progression (AP)

Terms change by adding or subtracting a constant difference ($d$).

$$d = T_2 - T_1 = T_3 - T_2$$
$$T_n = a + (n - 1)d$$
$$S_n = \frac{n}{2}[2a + (n - 1)d]$$
Geometric Progression (GP)

Terms change by multiplying by a constant ratio ($r$).

$$r = \frac{T_2}{T_1} = \frac{T_3}{T_2}$$
$$T_n = a r^{n-1}$$
$$S_n = \frac{a(1 - r^n)}{1 - r} \quad (r < 1)$$
$$S_\infty = \frac{a}{1 - r} \quad (|r| < 1)$$

2. Arithmetic Progressions (AP) Step-by-Step

An Arithmetic Progression (AP) is a sequence in which each term after the first is formed by adding a fixed constant called the common difference ($d$).

If the terms are $T_1, T_2, T_3, \dots$ then:

  • First term: $a = T_1$
  • Common difference: $d = T_2 - T_1$ (Note: $d$ can be negative if the sequence is decreasing!)
  • General $n^{\text{th}}$ term: $T_n = a + (n - 1)d$
  • Sum of the first $n$ terms: $S_n = \frac{n}{2}[2a + (n - 1)d]$ or $S_n = \frac{n}{2}(a + l)$ where $l$ is the last term.

3. Worked Past Paper Example: Arithmetic Progression

ECZ Past Paper Question

Given the arithmetic progression: $7, 11, 15, 19, \dots$

Find:

  1. The $15^{\text{th}}$ term.
  2. The sum of the first 20 terms.
Full Step-by-Step Solution:

Step 1: Identify given parameters:

First term $a = 7$
Common difference $d = 11 - 7 = 4$

Part (a): Find $T_{15}$

$$T_n = a + (n - 1)d$$ $$T_{15} = 7 + (15 - 1)(4)$$ $$T_{15} = 7 + (14)(4)$$ $$T_{15} = 7 + 56 = 63$$ Final Answer: $T_{15} = 63$ [2 Marks]

Part (b): Find $S_{20}$

$$S_n = \frac{n}{2}[2a + (n - 1)d]$$ $$S_{20} = \frac{20}{2}[2(7) + (20 - 1)(4)]$$ $$S_{20} = 10[14 + (19)(4)]$$ $$S_{20} = 10[14 + 76]$$ $$S_{20} = 10[90] = 900$$ Final Answer: $S_{20} = 900$ [3 Marks]

4. Geometric Progressions (GP) Step-by-Step

A Geometric Progression (GP) is a sequence in which each successive term is obtained by multiplying the previous term by a fixed non-zero number called the common ratio ($r$).

  • First term: $a = T_1$
  • Common ratio: $r = \frac{T_2}{T_1} = \frac{T_3}{T_2}$
  • $n^{\text{th}}$ term: $T_n = a r^{n-1}$
  • Sum of $n$ terms: $S_n = \frac{a(1 - r^n)}{1 - r}$ (convenient when $r < 1$) or $S_n = \frac{a(r^n - 1)}{r - 1}$ (convenient when $r > 1$).

5. The Sum to Infinity Condition ($|r| < 1$)

One of the most frequently asked questions in ECZ exams asks: "Find the sum to infinity of the progression."

Critical Examination Rule:

A geometric progression has a sum to infinity ($S_\infty$) if and only if the series is convergent, meaning:

$$-1 < r < 1 \quad \text{or} \quad |r| < 1$$

If $r = 2$ or $r = -3$, the terms become infinitely large and $S_\infty$ does not exist. When $|r| < 1$, as $n \to \infty$, $r^n \to 0$, simplifying the sum formula to:

$$S_\infty = \frac{a}{1 - r}$$

6. Worked Past Paper Example: Geometric Progression

ECZ Grade 12 Past Paper

The first three terms of a geometric progression are $64, 32, 16, \dots$

Find:

  1. The common ratio $r$.
  2. The $8^{\text{th}}$ term.
  3. The sum to infinity ($S_\infty$).
Full Step-by-Step Solution:

Part (a): Find the common ratio $r$

$$r = \frac{T_2}{T_1} = \frac{32}{64} = \frac{1}{2} \text{ (or } 0.5\text{)}$$ Final Answer: $r = \frac{1}{2}$ [1 Mark]

Part (b): Find the $8^{\text{th}}$ term ($T_8$)

$$T_n = a r^{n-1}$$ $$T_8 = 64 \times \left(\frac{1}{2}\right)^{8-1} = 64 \times \left(\frac{1}{2}\right)^7$$ $$\left(\frac{1}{2}\right)^7 = \frac{1}{128}$$ $$T_8 = 64 \times \frac{1}{128} = \frac{64}{128} = \frac{1}{2}$$ Final Answer: $T_8 = \frac{1}{2}$ (or $0.5$) [2 Marks]

Part (c): Find the sum to infinity ($S_\infty$)

Since $|r| = \left|\frac{1}{2}\right| < 1$, $S_\infty$ exists: $$S_\infty = \frac{a}{1 - r} = \frac{64}{1 - \frac{1}{2}} = \frac{64}{\frac{1}{2}} = 64 \times 2 = 128$$ Final Answer: $S_\infty = 128$ [2 Marks]

7. Top 4 Pitfalls That Cost Learners Marks

  • Incorrect Common Difference Sign: In a decreasing sequence like $20, 16, 12, \dots$, learners often write $d = 4$ instead of $d = 16 - 20 = -4$. Always compute $T_2 - T_1$.
  • Order of Operations in $T_n$: Writing $a + (n-1)d$ as $(a + n - 1)d$. Remember multiplication takes precedence over addition: multiply $(n-1)$ by $d$ first, then add $a$.
  • Power of Common Ratio: For $T_n = ar^{n-1}$, do NOT multiply $a$ by $r$ before taking the power! Calculate $r^{n-1}$ first, then multiply by $a$.
  • Sum to Infinity with $r > 1$: Attempting to use $S_\infty = \frac{a}{1 - r}$ when $r = 3$. This is mathematically invalid and awards zero marks.

8. Frequently Asked Questions

Can an AP also have a sum to infinity?

No. An AP continues to grow without bound either positively (if $d > 0$) or negatively (if $d < 0$). It does not have a finite sum to infinity unless all terms and $d$ are zero.

Can the common difference or common ratio be a fraction?

Yes. Both $d$ and $r$ can be positive or negative integers, fractions, or decimals. In ECZ exams, fractional values like $r = \frac{1}{3}$ or $d = -\frac{1}{2}$ are common.

Are progression formulas provided in the ECZ exam paper?

Yes, the front page of ECZ Mathematics Paper 2 provides formulas for $S_n$ and $S_\infty$, but you must know how to substitute into them accurately without copying errors.

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