SEQUENCES AND SERIES

GRADE 12

OUTCOMES OF LESSON:• Revision of linear and quadratic sequences.• An introduction to the arithmetic sequence and

formula.• An introduction to the geometric sequence and

formula.• Lastly , you will learn about Series for example both

arithmetic and geometric series.

REVISION OF QUADRATIC PATTERNS

• In grades 10 and 11 we dealt with quadratic number patterns of the general term +bn+c and linear or (arithmetic sequence) with the general term

• Remember a quadratic pattern has a second constant difference among consecutive terms and linear patterns a 1st constant difference.

EXAMPLECONSIDER THE FOLLOWING NUMBER PATTERN2 ; 3 ; 6 ; 11 ; …

1. Determine the and hence the value of the 42nd term.

2. Determine which term will equal 1091.

REVISION EXERCISE: DETERMINE THE GENERAL TERM FOR EACH NUMBER PATTERN.A.2 ; 6 ; 14 ; 26 ; …B.4 ; 9 ; 16 ; 25 ; …C.1; 3 ; 6 ; 10 ; …D.-1 ; 0 ; 3 ; 8 ; …E.-3 ; -6 ; -11 ; -18 ; …

ARITHMETIC SEQUENCES

• Note we can rewrite this pattern using only the first (1st) term and the Constant difference

• Consider the following linear number pattern: 7 ; 10 ; 13 ; 16 ; 19 ; …

• Term 1 T1=7 and constant difference = T2-T1=T3-T2=d=3

ARITHMETIC SEQUENCEST1=7T2=7+3=10T3=7+3+3=7+2(3)=13T4=7+3+3+3=7+3(3)=16THEREFORE, WE CAN CONCLUDE THAT T10=7+9(3)=34

ARITHMETIC SEQUENCES

• Therefore, the general term of the pattern is and we can therefore simplify this expression into

• In most cases and we can use these variables to write an expression for the general term that is .

EXAMPLE 1: CONSIDER THE FOLLOWING ARITHMETIC SEQUENCE 3 ; 5 ; 7 ; 9 ; …

a)Determine a formula for the general term of the above sequence.

b)Find the value of the 50th term.

EXAMPLE 2.CONSIDER THE FOLLOWING SEQUENCE: -5 ; -9 ; -13 ; -17 ; …

I. Show that the following sequence is arithmetic.

II.Find the value of the 25th term of the sequence.

EXERCISE 1a. are the first three terms of an arithmetic sequence.

Determine the value of x and hence the sequence.b.Determine the general term of the following linear

sequence 4 ; -2 ; -8 ; …c. are the first three terms of an arithmetic sequence.

Calculate the value of p. Moreover, determine the sequence and calculate the 49th term. Which term is equal to 100,5?

GEOMETRIC SEQUENCES: LET US TAKE A LOOK AT THE FOLLOWING NUMBER PATTERN 6 ; 12 ; 24 ; 48 ; 96 ; ….

• It is clear that this pattern is not linear (arithmetic) since there is no constant difference between consecutive terms. Neither , is there a second constant difference. However, this sequence is obtained by multiplying the previous term by 2 .

• Notice ,that we can calculate a ratio among consecutive terms in the following way =r

• This ratio we calculate is called the “constant ratio” and therefore is used to determine the next term by either multiplying or dividing by this constant ratio.

GEOMETRIC SEQUENCES: 6 ; 12 ; 24 ; 48 ; 96 ; …• This kind of number pattern is called a geometric or exponential pattern.

Moreover, we can rewrite this pattern using only the first term and the constant ratio.

• T1=6• T2=6×2=12• T3=6×2×2=6×24• T4=6×2×2×2=6×48• Therefore, T100=6× and therefore in algebraic terms and is known as the

general pattern of the number pattern.

GEOMETRIC SEQUENCES:

• If a is the first term and r is the constant ratio among consecutive terms of a geometric sequence then the pattern can be written as follows:

• T1=a• T2=ar• T3=aו T4=a• T100=a×