ACJC 2025 Sequences and Series Lecture Notes

1 4 SEQUENCES AND SERIES SYLLABUS ▪ Concepts of sequence and series for finite and infinite cases ▪ Sequence as function f ( )yn= where n is a positive integer ▪ Relationship between nu (the nth term) and nS (the sum to n terms) ▪ Sequence given by a formula for the nth term ▪ Sequence generated by a relation 1 f ( )nnuu+ = , including the use of a graphing calculator or a computer to generate the sequence ▪ Use of  notation ▪ Sum and difference of two series ▪ Convergence of a series and the sum to infinity ▪ Formula for the nth term and the sum of a finite arithmetic series ▪ Formula for the nth term and the sum of a finite geometric series ▪ Condition for convergence of an infinite geometric series ▪ Formula for the sum to infinity of a convergent geometric series

ACJC 2025/26 H2 Mathematics 2 CONTENTS 1 Introduction ...................................................................................... 3 1.1 Sequence ................................................................................. 3 1.2 Series ....................................................................................... 3 2 Arithmetic Progression ..................................................................... 4 2.1 Introduction to Arithmetic Progression................................... 4 2.2 Definition and Formula for the nth term of an Arithmetic Progression .............................................................................. 4 2.3 Formula for the Sum of an Arithmetic Progression ................ 6 3 Geometric Progression ..................................................................... 9 3.1 Introduction to Geometric Progression ................................... 9 3.2 Definition and Formula for the nth term of a Geometric Progression ............................................................................ 10 3.3 Formula for the sum of a Geometric Progression ................. 12 3.4 Sum to Infinity of a Geometric Progression ......................... 14 4 Summary of Arithmetic and Geometric Progressions .................... 17 5 Sequences ....................................................................................... 22 5.1 Sequences defined by a formula for the nth term ................. 22 5.2 Sequences generated by a relation 1 f ( )nnuu+ = ................... 23 5.3 Convergence of a sequence ................................................... 25 6 Series .............................................................................................. 30 6.1 Sigma Notation  .................................................................. 30 6.2 Useful Results on Series ....................................................... 33 6.3 Convergence of a Series ........................................................ 37 Annex A: Practice Questions on Sequences and Series........................... 40

4 Sequences and Series 3 LECTURE 1 Lesson Outline • Introduction to the concepts & notations of sequences and series • Arithmetic Progression: - Definition and formula for the nth term of an Arithmetic Progression - Formula for the sum of an Arithmetic Progression 1 INTRODUCTION 1.1 Sequence A sequence refers to a set of numbers written consecutively and is usually written as 1 2 3, , ,..., ,... nu u u u where nu is called the nth term of the sequence. Consider the following sequence of numbers. Fill in the missing numbers and give the general term. Sequence nth term 1, 2, 4, 8, 16, __, __, 128 12 , 1, 2, 3,..., 8n nun −== 1 1 1 1 11, , , , , ,23456 1 7 , 1 8 ,… nu = 1, 1, 2, 3, 5, 8, 13, __, __, … Relationship between terms is The first sequence is a finite sequence since it has a finite number of terms, whereas the other two are infinite sequences. A sequence can also be called a progression. 1.2 Series A series is formed when the terms of a sequence are added, i.e., 1 2 3 nu u u u+ + + + + . Examples 1. 2 4 8 16 32+ + + + is a finite series 2. 1 1 1 1 2 4 8 161+ + + + + is an infinite series

ACJC 2025/26 H2 Mathematics 4 We denote the sum of the first n terms of a sequence by 1 2 3nnS u u u u= + + + + . Example The sum of the first 4 terms of the sequence 2, 4, 8, 16, 32, … is 4 2 4 8 16 30S = + + + = . The nth term of a sequence nu is related to the sum of the first n terms of the sequence by 1n n nu S S −=− . Example For the sequence 2, 4, 8, 16, 32, …, 4 2 4 8 16S = + + + , 5 2 4 8 16 32S = + + + + . Therefore, the 5th term of the sequence is 5 5 4 32u S S= − = . In this chapter, we will study two special types of sequences and their series, namely, the arithmetic progression and the geometric progression. 2 ARITHMETIC PROGRESSION 2.1 Introduction to Arithmetic Progression The following three sequences follow a simple pattern. What are the next two terms? • 1, 4, 7, 10, __, __, … , 1,2,3,...nun== • −15, −11, −7, −3, __, __, … , 1,2,3,...nun== • 11 2229, 27 , 26, 24 , __, __, … , 1,2,3,...nun== 2.2 Definition and Formula for the nth term of an Arithmetic Progression An arithmetic progression (AP) is a sequence of numbers in which each term other than the first is obtained from the preceding one by the addition of a constant, called the common difference, d. The first number is called the first term, a. 1st term 2nd term 3rd term 4th term … nth term 1u 2u 3u 4u … nu a ad+ 2ad+ 3ad+ … ( 1)a n d+− Self-Practice Write down the general term nu

4 Sequences and Series 5 1. The formula for the nth term ( nu ) of an arithmetic progression is ( )1nu a n d= + − . 2. If 12, , ..., nu u u are consecutive terms of an arithmetic progression, then 1nnuu −− is a constant which is the common difference , d. Therefore 1nnd u u −=− . 3. To show that a sequence 12, , ... , nu u u is in arithmetic progression, we need to find nu and show that 1nnuu −− is a constant (i.e. 1nnuu −− is independent of n). It is insufficient to show that 21uu− is a constant. Example 1 Find the nth term of the arithmetic progression 15, 9, 3, … and hence find the 30th term. Solution AP with first term a = 15 and common difference d = nth term = 30th term 30u = ■ Example 2 The first four terms 1 2 3 4, , ,u u u u of an arithmetic progression are such that 42 15uu−= and 3149uu= . Find the value of 1u . Solution 42 15uu−= 3149uu= ■

ACJC 2025/26 H2 Mathematics 6 Example 3 The sum of the first n terms of a series is 223nS n n=− . (a) Write down the first term. (b) Find the nth term and show that the terms are in arithmetic progression. Solution (a) The first term 1u is (b) The nth term ■ 2.3 Formula for the Sum of an Arithmetic Progression Reference: https://nrich.maths.org/2478 Carl Friedrich Gauss (1777 -1855) is recognised as being one of the greatest mathematicians of all time. During his lifetime he made significant contributions to almost every area of mathematics, as well as physics, astronomy and statistics. Like many of the great mathematicians, Gauss showed amazing mathematical skill from an early age, and there are many stories which show how clever he could be. The most well-known story is a tale from when Gauss was still at primary school. One day Gauss' teacher asked his class to add together all the numbers from 1 to 100, assuming that this task would occupy them for quite a while. He was shocked when young Gauss, after a few seconds of thought, wrote down the answer 5050. The teacher couldn't understand how his pupil had calculated the sum so quickly in his head, but the eight year old Gauss pointed out that the problem was actually quite simple. He had added the numbers in pairs – the first and the last, the second and the second to last and so on, observing that 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so o