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Sequences and Series

Algebra · Sequences and Series

Syllabus tag: KCSE | Mathematics | Form 3 | Topic 10 Sequences and Series

Lesson objectives

By the end of this topic, you should be able to:

  • Identify number patterns and define a sequence.
  • Deduce the general rule for a given set of numbers.
  • Recognise arithmetic and geometric sequences and find their nth terms.
  • Find the sum of an arithmetic or geometric series.

Sequences and Series

A sequence is an ordered list of numbers following a rule. A series is what you get when you add them.

a) Finding the rule

Look at how each term relates to the one before.

A constant difference means arithmetic. A constant ratio means geometric.

Neither pattern? Look at squares, cubes, or differences of differences.

b) Arithmetic sequences

Arithmetic sequence Arithmetic sequence sequence = 3, 7, 11, 15, … constant difference first term a = 3 and d = 4 nth term = a + (n − 1)d the general rule 10th term = 3 + 9(4) substitute n = 10 10th term = 39 the answer

An arithmetic progression has a constant common difference d.

The nth term is a + (n − 1)d, where a is the first term.

The (n − 1) matters. Getting to the 10th term takes 9 steps, not 10.

c) Sum of an arithmetic series

Sum by pairing Sum by pairing the series = 1 + 2 + … + 100 Gauss's method pair the ends = 1+100, 2+99, 3+98 each pair gives 101 pairs = 50 half of 100 terms sum = 50 × 101 pairs times pair total sum = 5050 so Sₙ = n/2 (a + l)

Write the series forwards and backwards, then pair the terms. Every pair sums to the same value.

That gives Sₙ = n/2 (a + l), where l is the last term.

A second form follows. Since l = a + (n − 1)d, we also get Sₙ = n/2 [2a + (n − 1)d].

Use the first form when you know the last term, the second when you know d.

d) Geometric sequences

Geometric sequence Geometric sequence sequence = 2, 6, 18, 54, … constant ratio first term a = 2 and r = 3 nth term = arⁿ⁻¹ the general rule sum of n = a(rⁿ − 1)/(r − 1) when r is not 1

A geometric progression has a constant common ratio r. Find it by dividing any term by the one before.

The nth term is arⁿ⁻¹.

e) Sum of a geometric series

Sₙ = a(rⁿ − 1)/(r − 1) when r > 1.

The equivalent form a(1 − rⁿ)/(1 − r) is easier when r < 1, since it avoids negatives.

Both give the same answer. Choose whichever keeps the arithmetic positive.

f) Sum to infinity

If |r| < 1, the terms shrink towards zero and the sum approaches a limit.

S∞ = a/(1 − r).

Take 8 + 4 + 2 + 1 + …, where a is 8 and r is ½. The sum to infinity is 8 ÷ ½ = 16.

This only works when |r| < 1. If r is 1 or more, the terms do not shrink and the sum grows without limit.

g) Where this is used

Loan repayments and savings plans. Depreciation. Population models. Seating arrangements where each row grows by a fixed number.

Words to know

  • Sequence -- an ordered list of numbers.
  • Series -- the sum of the terms of a sequence.
  • Common difference -- the constant added in an arithmetic sequence.
  • Common ratio -- the constant multiplier in a geometric sequence.
  • Sum to infinity -- the limit of a geometric series when −1 < r < 1.

:::checkpoint Check yourself

  1. Find the 15th term of 5, 9, 13, …
  2. Find the sum of the first 20 terms of that sequence.
  3. Find the 6th term of 3, 6, 12, …
  4. Find the sum to infinity of 9 + 3 + 1 + … :::

Bridge to practice

The exercises begin with recognising patterns and finding rules, move through arithmetic and geometric nth terms and sums, and finish with the sum to infinity and applications. For every sequence, first subtract consecutive terms and then divide them, to establish which type you have.

Check yourselfPractise Sequences and Series — 10 questions →Next in MathematicsBinomial Expansion →