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Sequences (arithmetic sequences) — KCSE Mathematics

KCSE Mathematics · 115 practice questions · 4 syllabus objectives · 4 revision lessons

35 easy42 medium38 hard

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 an arithmetic sequence; find the first term and common difference from a sequence or given terms

Apply the formula for the nth term of an AP: Tₙ = a + (n–1)d, and find the sum using Sₙ = n/2(2a + (n–1)d)

Solve problems involving arithmetic progressions in real-life contexts (savings, salary increments, stacking)

Sequences (arithmetic sequences)

Revision Notes

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

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d). To identify an arithmetic sequence, check if the difference between each pair of consecutive terms is the same.

Steps to identify an arithmetic sequence:

  1. Write down the terms of the sequence.
  2. Subtract each term from the next to find the differences.
  3. If the differences are equal, it is an arithmetic sequence.

To find the first term (a) and common difference (d):

  • The first term is the initial term of the sequence.
  • The common difference is found by subtracting the first term from the second term.

Example: Consider the sequence: 3, 7, 11, 15.

  • First term (a) = 3.
  • Common difference (d) = 7 - 3 = 4.

Thus, the sequence is arithmetic with a first term of 3 and a common difference of 4.

Key points to remember

  • Arithmetic sequences have a constant difference between terms.
  • The first term is the initial term of the sequence.
  • The common difference is the difference between consecutive terms.
  • Identify a sequence by checking if differences are equal.

Worked example

Identify the first term and common difference of the sequence: 5, 10, 15, 20.

  • First term (a) = 5.
  • Common difference (d) = 10 - 5 = 5.

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More lessons in this topic

Lesson 2: Understanding Arithmetic Sequences

Objective: Apply the formula for the nth term of an AP: Tₙ = a + (n–1)d, and find the sum using Sₙ = n/2(2a + (n–1)d)

An arithmetic sequence (AP) is a sequence of numbers in which the difference between consecutive terms is constant. The formula for the nth term, Tₙ, is given by:

Tₙ = a + (n – 1)d

Where:

  • a = the first term
  • d = the common difference
  • n = the term number

To find the sum of the first n terms, we use:

Sₙ = n/2(2a + (n – 1)d)

This formula allows us to calculate the total of the first n terms in the sequence.

Example:

  • Given an AP where a = 3 and d = 5, find the 10th term and the sum of the first 10 terms.

Solution:

  1. Calculate T₁₀:
    • T₁₀ = 3 + (10 – 1)5 = 3 + 45 = 48
  2. Calculate S₁₀:
    • S₁₀ = 10/2(2(3) + (10 – 1)5) = 5(6 + 45) = 5 * 51 = 255

Thus, the 10th term is 48 and the sum of the first 10 terms is 255.

  • The nth term formula is Tₙ = a + (n – 1)d.
  • The sum formula is Sₙ = n/2(2a + (n – 1)d).
  • Identify a, d, and n to use the formulas correctly.
  • Arithmetic sequences have a constant difference between terms.

Find the 5th term and the sum of the first 5 terms for a sequence where a = 2 and d = 3. T₅ = 2 + (5 – 1)3 = 2 + 12 = 14; S₅ = 5/2(2(2) + (5 – 1)3) = 5/2(4 + 12) = 5/2(16) = 40.

Lesson 3: Understanding Arithmetic Progressions in Real Life

Objective: Solve problems involving arithmetic progressions in real-life contexts (savings, salary increments, stacking)

Arithmetic progressions (AP) are sequences where each term after the first is obtained by adding a constant difference. This concept is widely applicable in real-life scenarios such as savings, salary increments, and stacking items.

Key features of an arithmetic progression:

  • First term (a): The initial value in the sequence.
  • Common difference (d): The constant amount added to each term.
  • n-th term formula: The n-th term can be calculated using:
    [ a_n = a + (n - 1)d ]

Example 1: Savings
If you save Ksh 1,000 every month, your savings form an AP with:

  • First term (a) = 1,000
  • Common difference (d) = 1,000
    To find your savings after 12 months:
    [ a_{12} = 1000 + (12 - 1)1000 = 12,000 ]

Example 2: Salary Increment
If your salary increases by Ksh 5,000 each year, starting at Ksh 30,000, your salary after 5 years is:

  • First term (a) = 30,000
  • Common difference (d) = 5,000
    [ a_5 = 30000 + (5 - 1)5000 = 40000 ]
  • Arithmetic progressions have a constant difference between terms.
  • First term and common difference define the sequence.
  • The n-th term can be calculated using a specific formula.
  • Real-life applications include savings and salary increments.

If a person saves Ksh 2,000 monthly, calculate total savings after 10 months.

  • First term (a) = 2,000
  • Common difference (d) = 2,000
  • Total savings = a_{10} = 2000 + (10 - 1)2000 = 20,000.
Lesson 4: Understanding Arithmetic Sequences

Objective: Sequences (arithmetic sequences)

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the common difference (d). To describe an arithmetic sequence, we can use the formula for the nth term:

nth term (Tn) = a + (n - 1)d
Where:

  • a = first term
  • n = term number
  • d = common difference

For example, consider the sequence: 2, 5, 8, 11, ...

  • Here, the first term (a) is 2, and the common difference (d) is 3 (5 - 2).
  • To find the 5th term, substitute into the formula:
    • T5 = 2 + (5 - 1)3
    • T5 = 2 + 12 = 14.

Thus, the 5th term is 14. Remember, understanding the structure of arithmetic sequences helps in solving various mathematical problems effectively.

  • Arithmetic sequences have a constant difference between terms.
  • The formula for the nth term is Tn = a + (n - 1)d.
  • Identify the first term and common difference easily.

Find the 7th term of the sequence: 4, 9, 14, 19.

  • First term (a) = 4, common difference (d) = 5.
  • T7 = 4 + (7 - 1)5 = 4 + 30 = 34.

Sample Questions

Read 3 questions and answers free. Sign up to access all 115 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER2 marks

A certain savings plan starts with an initial deposit of KES 500 and increases by KES 100 each month. Identify the amount saved in the 6th month. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
T₆ = 500 + (6 - 1) × 100 (1 mk)
Amount saved in the 6th month is KES 1000 (1 mk)
2
easySHORT ANSWER4 marks

Calculate the sum of the first 15 terms of an arithmetic series where the first term is 2 and the common difference is 4. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Use the formula Sₙ = n/2(2a + (n - 1)d) (1 mk)
Substitute n = 15, a = 2, d = 4 into the formula (1 mk)
Calculate S₁₅ = 15/2(2*2 + (15 - 1)*4) (1 mk)
Find S₁₅ = 15/2(4 + 56) = 15/2(60) = 450 (1 mk)
3
easySHORT ANSWER3 marks

Explain how to find the 10th term of an arithmetic sequence where the first term is 5 and the common difference is 3. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Use the formula Tₙ = a + (n - 1)d (1 mk)
Substitute a = 5, n = 10, and d = 3 into the formula (1 mk)
Calculate T₁₀ = 5 + (10 - 1) * 3 = 32 (1 mk)
4

State the formula to find the sum of the first n terms of an arithmetic progression. (2 marks)

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Frequently asked questions

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

Sequences (arithmetic sequences) covers Identify an arithmetic sequence; find the first term and common difference from a sequence or given terms; Apply the formula for the nth term of an AP: Tₙ = a + (n–1)d, and find the sum using Sₙ = n/2(2a + (n–1)d); Solve problems involving arithmetic progressions in real-life contexts (savings, salary increments, stacking), and more, all aligned to the official KNEC KCSE Mathematics syllabus.

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

HighMarks has 115 Sequences (arithmetic 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 (arithmetic sequences) for the KCSE exam?

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