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Sequences (geometric sequences) — KCSE Mathematics

KCSE Mathematics · 114 practice questions · 4 syllabus objectives · 4 revision lessons

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Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Identify a geometric sequence; find the first term and common ratio from a sequence or given terms

Apply the formula for the nth term of a GP: Tₙ = arⁿ⁻¹, and find the sum using Sₙ = a(rⁿ – 1)/(r – 1)

Determine the sum to infinity of a convergent geometric series (|r| < 1) using S∞ = a/(1–r)

Sequences (geometric sequences)

Revision Notes

Concise lesson notes for Sequences (geometric sequences), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. To identify a geometric sequence, look for a consistent ratio between consecutive terms.

Steps to identify a geometric sequence:

  1. Check if the ratio between successive terms is constant.

  2. Find the first term, usually denoted as 'a'.

  3. Calculate the common ratio 'r' using the formula:

    r = an+1 / an

    where 'an' is the nth term and 'an+1' is the (n+1)th term.

Example:
Consider the sequence: 2, 6, 18, 54.

  • To find the common ratio:
    • r = 6 / 2 = 3
    • r = 18 / 6 = 3
    • r = 54 / 18 = 3
  • Since the ratio is constant, this is a geometric sequence.
  • The first term (a) is 2 and the common ratio (r) is 3.

Key points to remember

  • A geometric sequence has a constant ratio between terms.
  • Identify the first term as 'a'.
  • Calculate the common ratio as r = a<sub>n+1</sub> / a<sub>n</sub>.
  • Check multiple pairs of terms for consistency in ratio.
  • If ratios are the same, it is a geometric sequence.

Worked example

Identify the geometric sequence: 4, 12, 36, 108.

  • First term (a) = 4.
  • Common ratio (r) = 12 / 4 = 3.

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More lessons in this topic

Lesson 2: Understanding Geometric Sequences

Objective: Apply the formula for the nth term of a GP: Tₙ = arⁿ⁻¹, and find the sum using Sₙ = a(rⁿ – 1)/(r – 1)

In geometric sequences (G.P.), each term after the first is found by multiplying the previous term by a constant called the common ratio (r). The formula for the nth term (Tₙ) is given by:
Tₙ = arⁿ⁻¹
where:

  • a = the first term
  • r = the common ratio
  • n = the term number

To find the sum of the first n terms (Sₙ) of a geometric sequence, we use:
Sₙ = a(rⁿ – 1)/(r – 1), when r ≠ 1.

Example:
Given a G.P. where the first term (a) is 3 and the common ratio (r) is 2, find the 5th term and the sum of the first 5 terms.

  • Finding the 5th term:
    • T₅ = 3 × 2⁴ = 3 × 16 = 48
  • Finding the sum of the first 5 terms:
    • S₅ = 3(2⁵ – 1)/(2 – 1) = 3(32 – 1)/1 = 3 × 31 = 93
      Thus, the 5th term is 48 and the sum of the first 5 terms is 93.
  • Geometric sequences multiply by a common ratio.
  • The nth term formula is Tₙ = arⁿ⁻¹.
  • The sum formula is Sₙ = a(rⁿ – 1)/(r – 1).
  • Use these formulas to solve sequence problems.
  • Ensure r is not equal to 1 when using the sum formula.

Find the 4th term and the sum of the first 4 terms of a G.P. with a = 5 and r = 3.

  • T₄ = 5 × 3³ = 5 × 27 = 135
  • S₄ = 5(3⁴ – 1)/(3 – 1) = 5(81 – 1)/2 = 5 × 40/2 = 100.
Lesson 3: Sum to Infinity of Geometric Series

Objective: Determine the sum to infinity of a convergent geometric series (|r| < 1) using S∞ = a/(1–r)

In a convergent geometric series where the common ratio |r| is less than 1, the sum to infinity can be calculated using the formula:
S∞ = a / (1 - r)
Where:

  • S∞ is the sum to infinity.
  • a is the first term of the series.
  • r is the common ratio.
    To determine the sum to infinity, follow these steps:
  1. Identify the first term (a) of the series.
  2. Determine the common ratio (r).
  3. Ensure |r| < 1 for convergence.
  4. Substitute a and r into the formula.

Example:
Calculate the sum to infinity of the series:
4, 2, 1, ...

  • Here, a = 4 and r = 0.5 (since 2/4 = 0.5).
  • Check |r| < 1: Yes, |0.5| < 1.
  • Substitute into the formula:

S∞ = 4 / (1 - 0.5) = 4 / 0.5 = 8
Thus, the sum to infinity is 8.

  • Use S∞ = a / (1 - r) for convergence.
  • Identify the first term and common ratio.
  • Ensure |r| < 1 for the series to converge.
  • Substitute values correctly into the formula.

Find the sum to infinity of 3, 1.5, 0.75, ...

  • Here, a = 3, r = 0.5.
  • S∞ = 3 / (1 - 0.5) = 3 / 0.5 = 6.
Lesson 4: Understanding Geometric Sequences

Objective: Sequences (geometric sequences)

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. The general form of a geometric sequence can be expressed as:

a, ar, ar², ar³, ...

where:

  • a is the first term
  • r is the common ratio

To find the nth term of a geometric sequence, use the formula:

Tn = a * r^(n-1)

Example:
If the first term (a) is 3 and the common ratio (r) is 2, the sequence is:

  • 3, 6, 12, 24, ...
    To find the 5th term:
  • T5 = 3 * 2^(5-1)
  • T5 = 3 * 16
  • T5 = 48

Thus, the 5th term is 48.
Understanding geometric sequences is essential for solving problems in various mathematical contexts, including finance and growth models.

  • A geometric sequence has a constant ratio between consecutive terms.
  • The nth term formula is Tn = a * r^(n-1).
  • Identify the first term and common ratio for calculations.

Find the 4th term of the geometric sequence where a = 5 and r = 3.
T4 = 5 * 3^(4-1) = 5 * 27 = 135.

Sample Questions

Read 3 questions and answers free. Sign up to access all 114 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER3 marks

Identify the first term and common ratio of the geometric sequence where the third term is 24 and the sixth term is 192. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Let the first term be a and common ratio be r; then T3 = ar^2 = 24 (1 mk)
Also, T6 = ar^5 = 192 (1 mk)
Solving these equations gives a = 8 and r = 3 (1 mk)
2
easySHORT ANSWER3 marks

Name the first term and the common ratio of the geometric sequence where the second term is 12 and the fifth term is 96. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
First term (a) is 3 (1 mk)
Common ratio (r) is 4 (1 mk)
Correct identification of terms (1 mk)
3
easySHORT ANSWER4 marks

Identify the first term and common ratio of the geometric sequence where the 1st term is 5 and the 4th term is 40. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
First term is 5 (2 mks)
Part (b) — 2 marks
Common ratio is 2 (2 mks)
4

Name the first term and common ratio of the geometric sequence: 3, 6, 12, 24. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Sequences (geometric sequences)" cover?

Sequences (geometric sequences) covers Identify a geometric sequence; find the first term and common ratio from a sequence or given terms; Apply the formula for the nth term of a GP: Tₙ = arⁿ⁻¹, and find the sum using Sₙ = a(rⁿ – 1)/(r – 1); Determine the sum to infinity of a convergent geometric series (|r| < 1) using S∞ = a/(1–r), and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Sequences (geometric sequences)?

HighMarks has 114 Sequences (geometric sequences) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Sequences (geometric sequences) for the KCSE exam?

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