Arithmetic and Geometric Progressions

Q: Do I need to memorise AP/GP formulae for the 0606 exam?

No, all required AP/GP formulae are provided in the exam formula booklet for both Paper 1 and Paper 2. You only need to know when and how to apply them correctly.

Q: When can I use the sum to infinity formula for a GP?

You can only use if the common ratio satisfies (the GP is convergent). You must explicitly state this condition in exam answers to get full marks.

Additional Mathematics· 2025-2027 Syllabus, Sections 12.3, 12.4, 12.5· 25 min read

1. Identifying Arithmetic vs Geometric Progressions★☆☆☆☆⏱ 5 min

📘 Definition

Arithmetic Progression (AP)

$a, a + d, a + 2 d, a + 3 d, ...$

A sequence where consecutive terms differ by a constant common difference $d$ , which can be positive, negative, or zero.

Example:

2, 5, 8, 11, ... (d = 3)

📘 Definition

Geometric Progression (GP)

$a, a r, a r^{2}, a r^{3}, ...$

A sequence where each consecutive term is the previous term multiplied by a constant common ratio $r$ , which can be positive, negative, integer or fractional.

Example:

4, 2, 1, 0.5, ... (r = 1/2)

📐 Worked Example

Classify the following sequences as AP, GP, or neither: (a) 12, 7, 2, -3, ... (b) 3, 9, 27, 81, ... (c) 1, 4, 9, 16, ...

1
For (a): Calculate difference between consecutive terms: $7 - 12 = - 5$ , $2 - 7 = - 5$ , $- 3 - 2 = - 5$ . Constant difference = -5 → AP.
2
For (b): Calculate ratio between consecutive terms: $9/3 = 3$ , $27/9 = 3$ , $81/27 = 3$ . Constant ratio = 3 → GP.
3
For (c): Differences: $4 - 1 = 3$ , $9 - 4 = 5$ , $16 - 9 = 7$ (not constant). Ratios: $4/1 = 4$ , $9/4 = 2.25$ , $16/9 \approx 1.78$ (not constant) → neither.

✓ Quick check

Is the sequence 2, -4, 8, -16, ... an AP, GP, or neither?
- AP
- GP
- Neither
Reveal answer
GP —
Correct, the common ratio is -2, which is constant for all consecutive terms.
What is the common difference of the AP 10, 5, 0, -5, ...?
- 5
- 0
- -5
- 10
Reveal answer
-5 —
Correct, subtract consecutive terms: $5 - 10 = - 5$ , which is constant.

2. nth Term and Sum of First n Terms for AP★★☆☆☆⏱ 7 min

The nth term of an AP and sum of first n terms are provided in your exam formula booklet. For an AP with first term $a$ , common difference $d$ , and number of terms $n$ :

u_n = a + (n-1)d, \quad S_n = \frac{n}{2}[2a + (n-1)d] = \frac{n}{2}(a + l)

📐 Worked Example

An AP has first term 7 and common difference -2. Find (a) the 15th term, (b) the sum of the first 20 terms.

1
Identify given values: $a = 7$ , $d = - 2$ . For part (a), $n = 15$ .
2
Substitute into nth term formula:
$u_{15} = 7 + (15-1)(-2) = 7 - 28 = -21$
3
For part (b), $n = 20$ , substitute into sum formula:
$S_{20} = \frac{20}{2}[2(7) + (20-1)(-2)] = 10(14 - 38) = -240$

Exam tip:

When given the sum of terms, rearrange the sum formula to form a quadratic equation if needed, a common exam question style. Always state the values of a, d, n you are using to avoid arithmetic errors.

3. nth Term and Sum of First n Terms for GP★★★☆☆⏱ 7 min

For GPs, the nth term and finite sum formulae are also provided in the exam booklet. For a GP with first term $a$ , common ratio $r$ , and $n$ terms:

u_n = ar^{n-1}, \quad S_n = \frac{a(1 - r^n)}{1 - r} \quad (r \neq 1)

📐 Worked Example

A GP has first term 2 and common ratio 3. Find (a) the 6th term, (b) the sum of the first 8 terms.

1
Identify given values: $a = 2$ , $r = 3$ . For part (a), $n = 6$ .
2
Substitute into nth term formula:
$u_6 = 2 \times 3^{6-1} = 2 \times 243 = 486$
3
For part (b), $n = 8$ , use the sum formula for $r > 1$ :
$S_8 = \frac{2(3^8 - 1)}{3 - 1} = 6561 - 1 = 6560$

📐 Worked Example

The 3rd, 5th and 8th terms of an arithmetic progression are three consecutive terms of a geometric progression. The common difference of the AP is not zero. Find the common ratio of the GP.

1
Step 1: Write the three AP terms using $u_{n} = a + (n - 1) d$ :
$u_3 = a + 2d, \quad u_5 = a + 4d, \quad u_8 = a + 7d$
2
Step 2: Three consecutive GP terms satisfy (middle) $^{2}$ = (first)(last), so the middle term is the geometric mean:
$(a + 4d)^2 = (a + 2d)(a + 7d)$
3
Step 3: Expand both sides:
$a^2 + 8ad + 16d^2 = a^2 + 9ad + 14d^2$
4
Step 4: Cancel $a^{2}$ and collect terms:
$2d^2 = ad$
5
Step 5: Since $d \neq = 0$ , divide by $d$ to get $a = 2 d$ . Substitute back to find the three GP terms:
$u_3 = 4d, \quad u_5 = 6d, \quad u_8 = 9d$
6
Step 6: The common ratio is the ratio of consecutive terms:
$r = \frac{u_5}{u_3} = \frac{6d}{4d} = \frac{3}{2}$

✓ Quick check

A GP has $a = 16$ , $r = 1/2$ . Find $S_{4}$ .
Reveal answer
30 —
Correct: $S_{4} = 16 \times \frac{1 - ( 0.5 ) ^{4}}{1 - 0.5} = 30$

4. Sum to Infinity of a Convergent GP★★★★☆⏱ 6 min

📘 Definition

Convergent Geometric Progression

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

A GP where the absolute value of the common ratio is less than 1 ( $∣ r ∣ < 1$ ), meaning terms get progressively closer to zero, so the sum of an infinite number of terms is a fixed finite value.

📐 Worked Example

A GP has first term 18 and common ratio 1/3. (a) Justify that the GP is convergent. (b) Find its sum to infinity.

1
For part (a): Calculate $∣ r ∣ = ∣1/3∣ = 1/3 < 1$ , so the GP is convergent.
2
For part (b), substitute $a = 18$ and $r = 1/3$ into the sum to infinity formula:
$S_\infty = \frac{18}{1 - 1/3} = \frac{18}{2/3} = 27$

Exam tip:

Logarithms may be used to solve for $n$ in GP problems where you need to find the smallest $n$ such that $u_{n}$ or $S_{n}$ meets a given condition, which is allowed per the 0606 syllabus.

5. Common Pitfalls

Wrong move:

Confusing AP and GP formulae, using addition instead of multiplication for GP terms

Why:

Students mix up the structure of AP and GP sequences, leading to incorrect substitutions

Correct move:

Always confirm the progression type first, write down a, d/r explicitly before substituting into any formula

Wrong move:

Using the sum to infinity formula for a GP with |r| ≥ 1

Why:

Forgetting the convergence condition leads to invalid finite sum values for divergent GPs, losing method marks

Correct move:

Every time you use $S_{\infty}$ , first state and verify that |r| < 1 in your answer

Wrong move:

Using $n$ instead of $n - 1$ in the exponent for GP nth term

Why:

The first term is $a r^{0}$ , so the nth term has exponent n-1, not n, leading to off-by-one errors

Correct move:

Test your nth term formula with n=1: you should get the given first term a

Wrong move:

Rounding intermediate values when calculating GP terms or sums

Why:

Rounding early propagates errors, especially for non-integer values of r

Correct move:

Keep intermediate values in exact fractional or exponent form until the final step, then round as required

Wrong move:

Assuming all sequences are either AP or GP

Why:

Many exam questions include sequences that are neither to test your ability to verify progression type

Correct move:

Always check for constant difference (AP) or constant ratio (GP) before applying any AP/GP formulae

6. Quick Reference Cheatsheet

Progression Type	nth Term Formula	Sum of First n Terms	Sum to Infinity (if valid)
Arithmetic Progression (AP)	$u_{n} = a + (n - 1) d$	$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$	AP has no finite sum to infinity
Geometric Progression (GP)	$u_{n} = a r^{n - 1}$	$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ ( $r \neq = 1$ )	$S_{\infty} = \frac{a}{1 - r}$ , only valid if $∣ r ∣ < 1$

7. Frequently Asked

Do I need to memorise AP/GP formulae for the 0606 exam?

No, all required AP/GP formulae are provided in the exam formula booklet for both Paper 1 and Paper 2. You only need to know when and how to apply them correctly.

When can I use the sum to infinity formula for a GP?

You can only use $S_{\infty} = \frac{a}{1 - r}$ if the common ratio satisfies $∣ r ∣ < 1$ (the GP is convergent). You must explicitly state this condition in exam answers to get full marks.

Going deeper

official_resourceCIE 0606 2025-2027 SyllabusOfficial syllabus reference for this topic

What's Next

Now that you have mastered AP and GP core skills, you are ready to tackle combined series problems and word problems commonly tested in the CIE 0606 exam. Progressions are often paired with algebra and logarithms to solve for unknown values of n, d, or r, so reinforcing your algebra manipulation skills will help you tackle more complex exam questions efficiently. In your next study sessions, practice past paper questions that mix AP and GP concepts, as these are high-value questions that test your ability to distinguish between progression types and apply the correct formulae. You can also apply your series skills to practical scenarios like compound interest and population growth problems, which are common exam contexts.