Patterns, sequences and series Grade 12 Questions and Answers

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=t2-t1=t3-t2=tn-tn-1
  • Tn is the value of the term position n, so Tn=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=dwe 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 nth term formula in the sequence

Tn=a+(n-1)d

Tn=3+(n-1)4

Tn=3+4n-4

T=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

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

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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 nth 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. Tn=a+(n-1)

Tn=2+(n-1)4

Tn=2+4n-4

=4n-2

  1. T25=4(25)-2

=98

  1. 4n-2=46

4n=46+2

4n=48

n=48/4=12

  1. 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 nth term in the sequence
  4. Determine the value of the 15th
  5. Which term has a value of 302?
  6. Is 150 a term in the sequence ? Justify your answer fully

Solutions

  1. d=T-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

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

Tn=a+(n-1)d

=5+(n-1)3

= 5+3n-3

= 3n+2

  1. T15=3(15)+2

=47

  1. 3n+2=302

3n=302-2

3n=300

n=300/3

n=100

  1. 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

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Why passing Grade 12 is very important

Grade 12 ultimately prepares you for the real world out there, and most importantly matric is your ultimate gateway for varsity. You really need good marks in Grade 12 in order to pass well and study what you like. Good Grade 12 marks means good APS Score.

The calculation of an Admission Point Score (APS) is based on a candidate’s achievement in any SIX recognised National Senior Certificate 20-credit subjects (including subjects from the non-designated subject list, e.g. CAT, Tourism, Hospitality Studies and Civil, Electrical & Mechanical Technology, etc.), by using the NSC seven-point rating scale. Life Orientation is a 10-credit subject and may not be used for calculating the APS. Life Orientation is also not a faculty-specific subject requirement.

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5 Ways to Pass Matric with Distinction?

  1. Teach your friends the subject you know the most. Teaching others makes you learn faster
  2. Manage your time properly. Learn to focus at one task at a time.
  3. Gather all learning material about a subject and try answering previous exam papers and tests
  4. Ensure you do have 8 hrs of sleep every night so that you will always be productive.
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