Mathematics Form 1-4 : CHAPTER FIFTY – SEQUENCE AND SERIES

Specific Objectives

By the end of the topic, the learner should be able to:

• (a) Identify simple number patterns;
• (b) Define a sequence;
• (c) Identify the pattern for a given set of numbers and deduce the general rule;
• (d) Determine a term in a sequence;
• (e) Recognize arithmetic and geometric sequences;
• (f) Define a series;
• (g) Recognize arithmetic and geometric series (Progression);
• (h) Derive the formula for the partial sum of an arithmetic and geometric series (Progression);
• (i) Apply A.P and G.P to solve problems in real-life situations.

Content

• (a) Simple number patterns
• (b) Sequences
• (c) Arithmetic sequence
• (d) Geometric sequence
• (e) Determining a term in a sequence
• (f) Arithmetic progression (A.P)
• (g) Geometric progression (G.P)
• (h) Sum of an A.P
• (i) Sum of a G.P (excluding sum to infinity)
• (j) Application of A.P and G.P to real-life situations.

Introduction

Sequences and series are basically just numbers or expressions arranged in a row that form some sort of a pattern. For example, Monday, Tuesday, Wednesday, …, Friday is a sequence that represents the days of the week. Each of these numbers or expressions are called terms or elements of the sequence.

Sequences are the list of these items, separated by commas, and series are the sum of the terms of a sequence.

Example

Example

For the term of a sequence given by 2n + 3, find the first, fifth, and twelfth terms.

Solution

• First term, n = 1: 2 × 1 + 3 = 5
• Fifth term, n = 5: 2 × 5 + 3 = 13
• Twelfth term, n = 12: 2 × 12 + 3 = 27

Arithmetic and Geometric Sequence

Arithmetic sequence

Any sequence of numbers with a common difference is called an arithmetic sequence.

To decide whether a sequence is arithmetic, find the differences of consecutive terms. If each difference is constant, then it is an arithmetic sequence.

Rule for an arithmetic sequence

The nth term of an arithmetic sequence with first term a and common difference d is given by:

a_n = a + (n – 1)d

Example

Write a rule for the nth term of the sequence 50, 44, 38, 32, … Then find the 20th term.

Solution

The sequence is arithmetic with first term = 50 and common difference d = 44 – 50 = -6. So, a rule for the nth term is:

a_n = 50 + (n – 1)(-6)

Simplifying:

a_n = 56 – 6n

The 20th term is: 56 – 6(20) = -64.

Example

The 20th term of an arithmetic sequence is 60 and the 16th term is 20. Find the first term and the common difference.

Solution

• Subtracting the two terms: 20d = 40, so d = 10.

Therefore, a + 15 × 10 = 20

a + 150 = 20

a = -130

Hence, the first term is –130 and the common difference is 10.

Example

Find the number of terms in the sequence –3, 0, 3, … 54.

Solution

The nth term is given by a + (n – 1)d

• a = -3, d = 3
• nth term = 54
• Therefore, -3 + (n – 1) × 3 = 54
• 3(n – 1) = 57
• n – 1 = 19
• n = 20

Arithmetic series / Arithmetic progression (A.P)

The sum of the terms of a sequence is called a series. If the terms of a sequence are 1, 2, 3, 4, 5, when written with addition signs we get an arithmetic series:

1 + 2 + 3 + 4 + 5

The general formula for finding the sum of the terms is:

Note:

If the first term (a) and the last term (l) are given, then

Example

The sum of the first eight terms of an arithmetic progression is 220. If the third term is 17, find the sum of the first six terms.

Solution

S₈ = 4(2a + 7d)

So, 8a + 28d = 220 …………… 1

The third term is a + 2d = 17 …………… 2

Solving 1 and 2 simultaneously:

8a + 28d = 220 …………… 1

8a + 16d = 136 …………… 2

Subtracting gives 12d = 84

Substituting d = 7 in equation 2 gives a = 3

Therefore,

S₆ = 3(6 × 35)

= 3 × 41

= 123

Geometric sequence

It is a sequence with a common ratio. The ratio of any term to the previous term must be constant.

Rule for Geometric sequence:

The nth term of a geometric sequence with first term a and common ratio r is given by:

a_n = a × r^n-1

Example

Given the geometric sequence 4, 12, 36, … find the 4^th, 5^th, and nth terms.

Solution

The first term, a = 4

The common ratio, r = 3

Therefore, the 4^th term = 4 × 3³ = 4 × 27 = 108

The 5^th term = 4 × 3⁴ = 4 × 81 = 324

The nth term = 4 × 3^n-1

Example

The 4^th term of a geometric sequence is 16. If the first term is 2, find:

• The common ratio
• The seventh term

Solution

The first term, a = 2

The 4^th term is 2 × r³ = 16

Thus, r³ = 8

The common ratio is 2

The seventh term = 2 × 2⁶ = 2 × 64 = 128

Geometric series

The series obtained by adding the terms of a geometric sequence is called a geometric series or geometric progression (G.P).

The sum of the first n terms of a geometric series with common ratio r > 1 is:

S_n = a(rⁿ – 1) / (r – 1)

The sum of the first n terms of a geometric series with common ratio r < 1 is:

S_n = a(1 – rⁿ) / (1 – r)

Example

Find the sum of the first 9 terms of G.P. 8 + 24 + 72 + …

Solution

S₉ = 8(3⁹ – 1) / (3 – 1)

Example

The sum of the first three terms of a geometric series is 26. If the common ratio is 3, find the sum of the first six terms.

Solution

S₃ = a(3³ – 1) / (3 – 1) = 26

Find a and then calculate S₆.

End of topic

Past KCSE Questions on the topic

1. The first, the third, and the seventh terms of an increasing arithmetic progression are three consecutive terms of a geometric progression. If the first term of the arithmetic progression is 10, find the common difference of the arithmetic progression.
2. Kubai saved Ksh 2,000 during the first year of employment. In each subsequent year, he saved 15% more than the preceding year until he retired.
   • (a) How much did he save in the second year?
   • (b) How much did he save in the third year?
   • (c) Find the common ratio between the savings in two consecutive years.
   • (d) How many years did he take to save a sum of Ksh 58,000?
   • (e) How much had he saved after 20 years of service?
3. In a geometric progression, the first term is a and the common ratio is r. The sum of the first two terms is 12 and the third term is 16.
   • (a) Determine the ratio ar² / (a + ar).
   • (b) If the first term is larger than the second term, find the value of r.
4. (a) The first term of an arithmetic progression is 4 and the last term is 20. The sum of the terms is 252. Calculate the number of terms and the common difference of the arithmetic progression.
5. (b) An experimental culture has an initial population of 50 bacteria. The population increased by 80% every 20 minutes. Determine the time it will take to have a population of 1.2 million bacteria.
6. Each month, for 40 months, Amina deposited some money in a saving scheme. In the first month she deposited Kshs 500. Thereafter she increased her deposits by Kshs 50 every month.
   • (a) Last amount deposited by Amina
   • (b) Total amount Amina had saved in the 40 months.
7. A carpenter wishes to make a ladder with 15 cross-pieces. The cross-pieces are to diminish uniformly in length from 67 cm at the bottom to 32 cm at the top. Calculate the length in cm of the seventh cross-piece from the bottom.
8. The second and fifth terms of a geometric progression are 16 and 2 respectively. Determine the common ratio and the first term.
9. The eleventh term of an arithmetic progression is four times its second term. The sum of the first seven terms of the same progression is 175.
   • (a) Find the first term and common difference of the progression.
   • (b) Given that the p^th term of the progression is greater than 124, find the least value of p.
10. The nth term of a sequence is given by 2n + 3.
    • (a) Write down the first four terms of the sequence.
    • (b) Find S_n, the sum of the first fifty terms of the sequence.
    • (c) Show that the sum of the first n terms of the sequence is given by S_n = n² + 4n. Hence or otherwise find the largest integral value of n such that S_n < 725.