Patterns, sequences and series Grade 12 Questions and Answers

Patterns, sequences and series Grade 12 Questions and Answers:

Arithmetic sequences

We define an arithmetic sequence as follows:

a , a+d; a+2d; a+3d; a+4d; a+(n-1)d

• a the value of the first term
• d is the common difference between terms, d=t₂-t₁=t₃-t₂=t_n-t_n-1
• T_n is the value of the term position n, so T_n=a+(n-1)d
• n is the position of a term and can only be positive whole number, also known as a natural number

An arithmetic sequence is a sequence where consecutive terms are calculated by adding a constant value (positive or negative) to the previous term. We call this constant value the common difference (dd).

For example,3;0;−3;−6;−9;…

Consider the arithmetic sequence 3; 7; 11;15;…….99

T₁=3, T₂=7, T₃=11

d₁=T₂-T₁=7-3=4 , d₂=T₃-T₂=11-7=4, d₃=T₄-T₃=15-11=4

Since d₁=d₂=d₃ we have a common difference of 4

The first term is given by a=3 and the common difference is given by d=4

We use a and d to determine the n^th term formula in the sequence

T_n=a+(n-1)d

T_n=3+(n-1)4

T_n=3+4n-4

T_n­=4n-1

To prove that the formula is correct, you can check the formula by substituting n=1 to obtain the value of T₁ and n=2 to obtain the value of T₂ and so on

If n =1, then T₁=4(1)-1=3

If n=2, then T₂=4(2)-1=7

If n=3, then T₃=4(3)-1=11

T_n=4n-1 is a correct formula and can be used to determine the position of any term in the sequence

Worked Example

Consider the arithmetic sequence 2; 6; 10;14;………..

1. What is the common difference
2. State the values of the next two terms in the sequence
3. Determine a formula for the n^th term of the sequence
4. Determine the value of the of the twenty fifth term
5. Which term has a value of 46?
6. Is 72 a term in the sequence? Justify your answer.

Solutions

1. d₁=T₂-T₁=6-2=4
2. The common difference is 4, so the next two terms are 14+4=18 and 18+4=22
3. T_n=a+(n-1)

T_n=2+(n-1)4

T_n=2+4n-4

=4n-2

4. T₂₅=4(25)-2

=98

5. 4n-2=46

4n=46+2

4n=48

n=48/4=12

6. 4n-2=72

4n=72+2

4n=74

n=74/4=18.5

72 is not a term in this sequence because n is not a whole number.

Worked Example 2

Consider the arithmetic sequence 3x-1; 5x-2; 4x+3

1. Determine the value of x
2. If x=2 determine the values of the first three terms in the sequence
3. Determine the n^th term in the sequence
4. Determine the value of the 15^th
5. Which term has a value of 302?
6. Is 150 a term in the sequence ? Justify your answer fully

Solutions

1. d=T_2­-T₁=T₃-T₂

(5x-2)-(3x-1)=(4x+3)-(5x-2)

5x-2-3x+1=4x+3-5x+2

5x-3x-4x+5x=3+2+2-1

3x=6

X=6/3=2

2. T₁=3(2)-1=5, T₂=5(2)-2=8 and T₃=4(2)+3=11
3. a= 5, d= 8-5=3

T_n=a+(n-1)d

=5+(n-1)3

= 5+3n-3

= 3n+2

4. T₁₅=3(15)+2

=47

5. 3n+2=302

3n=302-2

3n=300

n=300/3

n=100

6. 3n+2=150

3n=150-2

3n=148

n=148/3

n is not a whole number ,therefore 150 is not a term in this sequence

Sources

• https://www.mathsman.co.za/grade-12-maths.html
• https://afrimaths.co.za/lesson/grade-12-lesson-1-patterns-sequences-series/
• https://intl.siyavula.com/read/maths/grade-12/sequences-and-series/01-sequences-and-series-01

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