Patterns, sequences and series Grade 12 Questions and Answers:

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### 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=t
_{2}-t_{1}=t_{3}-t_{2}=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

T_{1}=3, T_{2}=7, T_{3}=11

d_{1}=T_{2}-T_{1}=7-3=4 , d_{2}=T_{3}-T_{2}=11-7=4, d_{3}=T_{4}-T_{3}=15-11=4

Since d_{1}=d_{2}=d_{3 }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 T_{1} and n=2 to obtain the value of T_{2} and so on

If n =1, then T_{1}=4(1)-1=3

If n=2, then T_{2}=4(2)-1=7

If n=3, then T_{3}=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;………..

- What is the common difference
- State the values of the next two terms in the sequence
- Determine a formula for the n
^{th}term of the sequence - Determine the value of the of the twenty fifth term
- Which term has a value of 46?
- Is 72 a term in the sequence? Justify your answer.

Solutions

- d
_{1}=T_{2}-T_{1}=6-2=4 - The common difference is 4, so the next two terms are 14+4=18 and 18+4=22
- T
_{n}=a+(n-1)

T_{n}=2+(n-1)4

T_{n}=2+4n-4

=4n-2

- T
_{25}=4(25)-2

=98

- 4n-2=46

4n=46+2

4n=48

n=48/4=12

- 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

- Determine the value of x
- If x=2 determine the values of the first three terms in the sequence
- Determine the n
^{th}term in the sequence - Determine the value of the 15
^{th} - Which term has a value of 302?
- Is 150 a term in the sequence ? Justify your answer fully

Solutions

- d=T
_{2}-T_{1}=T_{3}-T_{2}

(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

- T
_{1}=3(2)-1=5, T_{2}=5(2)-2=8 and T_{3}=4(2)+3=11 - a= 5, d= 8-5=3

T_{n}=a+(n-1)d

=5+(n-1)3

= 5+3n-3

= 3n+2

- T
_{15}=3(15)+2

=47

- 3n+2=302

3n=302-2

3n=300

n=300/3

n=100

- 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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**Teach your friends**the subject you know the most. Teaching others makes you learn faster- Manage your time properly. Learn to
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