Mathematics · Lesson 12

AP and GP

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: sequences and series, nth term and sum of AP and GP, sum to infinity

WAEC: Sequences and series: patterns of sequences, nth term, arithmetic progression (AP) and geometric progression (GP), sum of AP and GP, sum to infinity of a GP, arithmetic and geometric means, word problems

NECO: Sequences and series: patterns of sequences, nth term, arithmetic progression (AP) and geometric progression (GP), sum of AP and GP, sum to infinity of a GP, arithmetic and geometric means, word problems

What you will learn

Recognise and continue number patterns; find a rule for the nth term.
Identify APs and GPs and find d or r.
Find any term and the number of terms.
Calculate the sum of n terms of an AP and GP.
Find the sum to infinity of a convergent GP; convert recurring decimals to fractions.
Insert arithmetic and geometric means.
Solve word problems on salaries, savings, population and depreciation.

Key points

A sequence follows a rule; a series is the sum of its terms.
AP: constant difference d. Tₙ = a + (n − 1)d.
AP sum: Sₙ = n/2[2a + (n − 1)d] = n/2(a + l).
GP: constant ratio r. Tₙ = arⁿ⁻¹.
GP sum: Sₙ = a(rⁿ − 1)/(r − 1) = a(1 − rⁿ)/(1 − r).
Sum to infinity exists only if |r| < 1: S∞ = a/(1 − r).
Recurring decimals are GPs: 0.272727… = 27/99 = 3/11.
x, y, z in AP: 2y = x + z. In GP: y² = xz.
AM = (a + b)/2; GM = √(ab).
Fixed amount change → AP; fixed percentage change → GP.

Start here: Sequences and patterns

A sequence is a list of numbers in a definite order that follows a rule. Each number is a term. We write T₁ for the first term, T₂ for the second, and Tₙ for the nth term.

Examples: 2, 4, 6, 8, … (add 2); 1, 4, 9, 16, … (square numbers); 1, 1, 2, 3, 5, 8, … (each term is the sum of the previous two, the Fibonacci sequence). Patterns are everywhere: the number of seats in successive rows of a stadium, or the number of matches in a growing pattern of squares.

The nth term (general term) is a formula giving Tₙ in terms of n. With it, you can find any term without listing all the ones before.

Worked example 12.1

Write the first four terms of Tₙ = 3n − 1.

n = 1: 2; n = 2: 5; n = 3: 8; n = 4: 11.

2, 5, 8, 11

Worked example 12.2

Find the nth term of 5, 9, 13, 17, …

The difference is 4 each time, so Tₙ starts with 4n.
When n = 1, 4n = 4, but T₁ = 5: add 1.
Tₙ = 4n + 1. Check: n = 3 gives 13 ✓.

Tₙ = 4n + 1

Worked example 12.3

Find the nth term of 2, 6, 12, 20, 30, …

Differences: 4, 6, 8, 10 (not constant), so try a product.
1×2, 2×3, 3×4, 4×5, 5×6.
Tₙ = n(n + 1).

n(n + 1)

⚠️ Common mistake: writing Tₙ = n + 4 for 5, 9, 13, … because "you add 4". The difference becomes the coefficient of n.

Try it yourself. Find the 10th term of Tₙ = n² − 2n.

Show solution

100 − 20 = 80.

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Practice questions

1. Find the 25th term of 8, 5, 2, …

Show worked answer

8 + 24(−3) = −64.

2. How many terms are in 7, 11, 15, …, 103?

Show worked answer

(103 − 7)/4 + 1 = 25.

3. Find the sum of the first 15 terms of 2, 5, 8, …

Show worked answer

7.5(4 + 42) = 345.

4. The 3rd and 6th terms of a GP are 12 and 96. Find the first term.

Show worked answer

r³ = 8, r = 2, a = 3.

5. Find the sum of the first 7 terms of 1, 2, 4, …

Show worked answer

127.

Key words

Sequence: An ordered list of numbers formed by a rule.
Term: Each number in a sequence; Tₙ is the nth term.
Series: The sum of the terms of a sequence.
Arithmetic progression (AP): A sequence with a constant difference between terms.
Common difference (d): The amount added each time in an AP.
Geometric progression (GP): A sequence with a constant ratio between terms.
Common ratio (r): The factor multiplied each time in a GP.
First term (a): The starting term T₁.
Last term (l): The final term of a finite sequence.
Sum to infinity: The limit of Sₙ as n becomes very large, for a convergent GP.
Arithmetic mean: The middle term between two numbers in an AP.
Geometric mean: The middle term between two numbers in a GP.

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