Further Mathematics · Lesson 7

Sequences & series: AP, GP, sum to infinity

WAECNECO

What the syllabus asks

WAEC: Sequences and series: finite and infinite sequences, linear (arithmetic) and exponential (geometric) sequences, nth term, sum of n terms, arithmetic and geometric means, sum to infinity of a convergent GP, recurring decimals, sigma notation, practical applications

NECO: Sequences and series: finite and infinite sequences, linear (arithmetic) and exponential (geometric) sequences, nth term, sum of n terms, arithmetic and geometric means, sum to infinity of a convergent GP, recurring decimals, sigma notation, practical applications

What you will learn

Find the rule, nth term and missing terms of sequences.
Solve AP problems involving nth term and sum of n terms.
Solve GP problems involving nth term and sum of n terms.
Find the sum to infinity of a convergent GP and convert recurring decimals to fractions.
Insert arithmetic and geometric means.
Use sigma notation and apply sequences to real problems (salaries, savings, depreciation, bouncing balls).

Key points

AP: Tₙ = a + (n − 1)d; Sₙ = n/2[2a + (n − 1)d] = n/2(a + l).
GP: Tₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1) or a(1 − rⁿ)/(1 − r).
S∞ = a/(1 − r) only if |r| < 1.
Three terms in AP: 2y = x + z. In GP: y² = xz.
AM = (x + y)/2, GM = √(xy); AM ≥ GM for positives.
Recurring decimals are infinite GPs: 0.4̇5̇ = 45/99 = 5/11.
Subtract term equations for AP, divide for GP; n must be a positive integer.

Start here: Sequences and their rules

A sequence is a list of numbers in a definite order, formed by a rule. For example, 2, 4, 6, 8, … (add 2 each time) or 1, 4, 9, 16, … (square numbers).

We write the terms as T₁, T₂, T₃, …, and the general term or nth term Tₙ is a formula that gives any term from its position n.

Tₙ = 2n gives 2, 4, 6, 8, …
Tₙ = n² gives 1, 4, 9, 16, …
Tₙ = 3n − 1 gives 2, 5, 8, 11, …
Tₙ = (−1)ⁿ gives −1, 1, −1, 1, … (alternating)

A series is what you get when you add terms: 2 + 5 + 8 + 11 is a series. Its sum S₄ = 26.

Worked example 7.1

Find the first three terms and the 10th term of Tₙ = n² + 2n.

Substitute n = 1, 2, 3: 1 + 2 = 3; 4 + 4 = 8; 9 + 6 = 15. We substitute because Tₙ is a formula in n.
T₁₀ = 100 + 20 = 120.

3, 8, 15; T₁₀ = 120

Worked example 7.2

Find the nth term of 5, 9, 13, 17, …

The difference is constant, 4, so Tₙ = 4n + c.
When n = 1, T₁ = 5: 4 + c = 5, so c = 1.

Tₙ = 4n + 1

Try it yourself. Find the 6th term of Tₙ = 2ⁿ − 1.

Show solution

2⁶ − 1 = 63.

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Practice questions

1. Find the 25th term of 7, 11, 15, …

Show worked answer

7 + 24 × 4 = 103.

2. Find the sum of the first 30 terms of 1 + 4 + 7 + …

Show worked answer

15[2 + 87] = 1,335.

3. The 4th term of a GP is 54 and the 7th is 1,458. Find a and r.

Show worked answer

r³ = 27, r = 3; a = 54/27 = 2.

4. Find S∞ of 50 + 10 + 2 + …

Show worked answer

50/(4/5) = 62.5.

5. Express 0.3̇6̇ as a fraction.

Show worked answer

36/99 = 4/11.

Key words

Sequence: An ordered set 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): Sequence with a constant difference between consecutive terms.
Common difference (d): Tₙ₊₁ − Tₙ in an AP.
Geometric progression (GP): Sequence with a constant ratio between consecutive terms.
Common ratio (r): Tₙ₊₁ ÷ Tₙ in a GP.
Finite sequence: One with a last term.
Infinite sequence: One that continues without end.
Convergent series: An infinite series whose sum approaches a fixed value.
Divergent series: An infinite series whose sum grows without limit.
Arithmetic mean: The middle term of three in AP: (x + y)/2.

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