RI APGP C7B Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 1 of 18 Chapter 7B: Arithmetic Progression and Geometric Progression SYLLABUS INCLUDES  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 CONTENT 3 Special Sequence 1: Arithmetic Progression (AP) 3.1 Definition 3.2 thn term of an Arithmetic Progression 3.3 Sum of first n terms of an Arithmetic Series 3.4 Prove that a Sequence is an Arithmetic Progression 4 Special Sequence 2: Geometric Progression (GP) 4.1 Definition 4.2 thn term of a Geometric Progression 4.3 Sum of first n terms of a Geometric Series 4.4 Prove that a Sequence is a Geometric Progression 4.5 Sum to infinity of a Geometric Series 5 Miscellaneous

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 2 of 18 INTRODUCTION One common type of sequence is the arithmetic progression (AP)—a sequence in which each term differs from the previous one by a constant amount. For example, saving a fixed amount of money every month or increasing your jogging distance by the same amount each week are everyday situations that involve an AP. Another important type is the geometric progression (GP), where each term is obtained by multiplying the previous one by a fixed number. GPs appear in many real-world settings too— such as when your savings grow with compound interest, or when a population doubles every few years. By understanding APs and GPs, you will gain useful tools for recognising patterns, predicting future values, and solving problems that pop up in both personal and financial planning. Let’s explore how these special sequences work—and why they matter! 3 SPECIAL SEQUENCE 1: ARITHMETIC PROGRESSION (AP) 3.1 Definition An arithmetic progression (AP) is a sequence of numbers for which the difference between every two consecutive terms is a constant, i.e. . The constant d is called the common difference. Note that we can easily see that AP is a sequence generated by a recurrence relation by writing it as 1nnuu d  . 3.2 n th term of an Arithmetic Progression An arithmetic progression with first term a and common difference d can be written as , , 2 , 3 , . . .aa da da d  The thn term of the arithmetic progression is (1 )nua n d  1nnuu d 

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 3 of 18 Examples Arithmetic Progression First Term 1u Common Difference Formula for nu 3, 5, 7, 9, 11, … 3 2 2 1, 1, 2,3,...nunn  20, 15, 10, 5, 0, … , 30 20 5  20 ( 1) 5 25 5 , 1, 2,3,...11 nun nn    357, 2 , , 3 , , . . .222 3 2 1 2 31 (1 ) 1 ,22 2 1, 2,3,... n nun n      An arithmetic series is formed when the terms of an arithmetic progression are added. For example, 3 5 7 9 11 ...  3.3 Sum of first n terms of an Arithmetic Series We illustrate with a simple example the method to find the sum of 1 2 3 ... 8 9 10 . 10 10 1 2 3 ... 8 9 10 (1) 10 9 8 ... 3 2 1 (2) S S          (1) + (2) gives  10 10 10 2 11 11 ... 11 21 0 1 1 110 552 S S S    Let nS denote the sum of the first n terms of an arithmetic series with first term a and common difference d .Then 123 2 1 ... ( ) ( 2 ) ... [ ( 3) ] [ ( 2) ] [ ( 1) ] (1) nn n nSu u u u u u aa d a d an d an d an d                 Similarly, 12 3 2 1 ... [ ( 1) ] [ ( 2) ] [ ( 3) ] ... ( 2 ) ( ) (2) nn n nSu u u u u u an d an d an d a d a da             (1) + (2) gives  22 ( 1 ) 2 ( 1 ) 2 nn nS n an d S an d    

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 4 of 18 Since (1 )nua n d  , we can also write nS as    12(1 ) (1 )22 2 n n nn nS a nd a a nd u u      Sum of the first n terms of an arithmetic series is  1[2 ( 1) ]22 nn nnSa n d u u   Remarks: Sum of first n terms of an arithmetic series with a as its first term and d as its common difference can be expressed as  1 (1 ) n k ak d   , which involves a linear expression in terms of k. We can apply the sum of first n terms formula of an arithmetic series on summation problems involving a general linear term. To illustrate this, let us revisit Example 5a. 10 0 (2 3) 3 5 7 ... 23 (Arithmetic Series with 1st term 3 and common difference 3) r r      10 0 1 (3 23) 1432   This method is more efficient than splitting the series as 10 10 10 00 0 (2 3) 2 3 rr r rr      which involves application of 2 formulae from Section 1.6 Summation using Standard Results: (1) 1 n r an a   . (2) 1 1 (1 )2 n r rn n   . Note that 1 n r r   is also an arithmetic series with n terms, first term 1 and common difference 1. We can easily apply the sum of first n terms formula to derive that 1 1 (1 )2 n r rn n   .

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 5 of 18 Example 9 Find the number of terms in the arithmetic progression 20, 15, 10, … , 135 and the sum of all the terms. Solution AP : first term a 20 , common difference d  5 , number of terms  n (1 )nua n d  135 20 ( 1)( 5)n    1551 5 32 n n   32 212 32[2(20) (32 1)( 5)]2 1840 nSa n d      OR 32 1 2 32 [20 ( 135)]2 1840 n nSu u    Example 10 In an arithmetic progression, the 8th term is twice the 4th term, and the 20th term is 40. Find the common difference and the sum of the terms from the 8th to the 20th inclusive. Solution AP : first term a , common difference d 84 2 7 2( 3 ) uu a da d a d     20 40 19 40 20 40 2ua d d a d      Sum of the terms from the 8th to the 20th inclusive  20 7SS   20 72(2) 19(2) 2(2) 6(2) 36422    Alternatively, we can consider the terms from the 8th to the 20th as a new AP with 8 20 1st term, 2 7(2) 16 last term, 2 19(2) 40 u u     Number of terms 20 8 1 13.    82 0 13 13Required sum 16 4022 364 uu  

Raffles Institution H2 Mathematics 2025 Year 5 _________________________________________________________________________________________ ______________________________ Chapter 7B: Arithmetic Progression and Geometric Progression Page 6 of 18 Example 11 An arithmetic progression whose first term is 2 and whose thn term is 32 has the sum of its first n terms equal to 357. Find n . If the thk term has a value exceeding 100, find the least value of k . Solution AP: first term 2a , common difference d 357 (2 32) 357 2 n nS   21n 21 32 2 20 32 nuu d   3 2d Find the least value of k such that 32( 1 ) 1 0 02 kuk  196 1 3 199 1 66 33 k k     Least value of k is 67.