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Grade 12 Number Patterns Prep Exam paper 1
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NUMBER PATTERNSGRADE 12
NOTES + ACTIVITIES+
PAST EXAM PAPERS
EMAILBY
ADDRESS: melulekishabalala@gmail.com
MR M. SHABALALA CELLPHONE NUMBER: 0733318802 Page 1
@NOMBUSO HIGH
, NUMBER PATTERNS SUMMARY
PATTERNS/SEQUENCES
1.Arithmetic sequence 2.Quadratic sequence 3.Geometric sequence
Common 1st difference Common 2nd difference Common ratio
Example Example Example
-1, 1, 3, 5 3, 13, 31, 57, 91 1, 3, 9, 27
\/ \/ \/ \/ \/ \/ \/ \/ \/ \/
2 2 2 common diff (d) = 2 10, 18, 26, 34 1st difference 3 9 27
\/ \/ \/
= = = 3 common ratio
1 3 9
∴T2 -T1 = T3 – T2 8 8 8 2nd difference
𝑻𝒏 = 𝒂 + (𝒏 − 𝟏)𝒅 𝑻𝒏 = 𝒂𝒏𝟐 + 𝒃𝒏 + 𝒄 𝑻𝒏 = 𝒂𝒓𝒏−𝟏
𝑎 →1st term r = common ratio
SERIES
(When we add terms of sequences together we form series/ (sum), usually terms are separated by +)
1. Arithmetic series 2.Geometric series
𝑺𝒏 = 𝒏𝟐 [𝟐𝒂 + (𝒏 − 𝟏)𝒅] 𝒂(𝒓𝒏 −𝟏)
𝑺𝒏 =
𝒓−𝟏
𝑺𝒏 = 𝒏𝟐 [𝒂 + 𝑳] only if last term is known Where
Where 𝑺𝒏 → is the sum of terms
𝑺𝒏 → is the sum of terms 𝒂 → is the 1st term
𝒂 → is the 1st term 𝒓 → is common ratio
𝒅 → is the common difference 𝒏 → is the term number
𝑳 → is the last term
𝒏 → is the term number
SIGMA NOTATION
Another useful way to represent a SERIES.
∑(sigma) denotes summation (addition)
Evaluate in sigma notation. 1st always expand to decide if it is arithmetic or geometric
Reads as: The sum of all terms(5𝑘 − 3) from 𝑘 = 4 to 𝑘 =7
number of terms = (top number – bottom number. ) + 1 Example 𝑘 = (7 − 4) + 1 = 4
SUM TO INFINITY
1.Calculations 2.Converge
𝒂
𝑺∞ = need only 1st term and ratio If sequence converge look at the common ratio
𝟏− 𝒓 −1< 𝑟 < 1
Need common ratio to decide
BY MR M. SHABALALA @NOMBUSO HIGH Page 2
, NUMBER PATTERNS GRADE 12
1. SEQUENCES (Arithmetic, Quadratic and Geometric)
Given three terms : (2𝑝 − 3); (𝑝 + 5) ; (2𝑝 + 7)
1.1 What type of pattern is this?
.
Is it Arithmetic? If it is arithmetic must have the following properties
ARITHMETIC SEQUENCE
Common difference FORMULAR:
( 𝑻𝟐 − 𝑻𝟏 = 𝑻𝟑 − 𝑻𝟐 ) 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
Is it Quadratic? If it is quadratic must have the following properties.
QUADRATIC SEQUENCE
Common FORMULAR: Steps to determine equation
second 𝟐
𝑻𝒏 = 𝒂𝒏 + 𝒃𝒏 + 𝒄 2𝑎 = 2nd common difference
difference 3𝑎 + 𝑏 = 1st common difference
𝑎 + 𝑏 + 𝑐 = 1st term
1.1 ANSWER: You can’t decide type of sequence when given variables, you need actual values
so that 1st difference for arithmetic/ 2nd different for quadratic maybe calculated
: You can’t solve variables if you are not told type of sequence.
1.2. Given the finite arithmetic sequence: −5 ; −1 ; 3 ; 7 ; … … … … … … … . ; 259
1.2.1 Write down the values of the next two terms
1.2.2 Determine the formula of the general term of the sequence
1.2.3 Calculate the value of the 31st term.
1.2.4 Determine the number of terms in the sequence.
1.2.5 Calculate the sum of the first 15 terms.
1.3 Given the following quadratic sequence: −2 ; 0 ; 3 ; 7; … …
1.3.1 Write down the values of the next two terms
1.3.2 Determine 𝑇246
Why is important to start with general formula to answer this question?
When to determine nth term or formula/formula?
1. Determine formula when asked in the question.
2. Determine formula when asked term of big number, if less, use manual count.
1.3.3 Which term of the sequence will be equals to 322?
1.3.4 Between which two consecutive terms of the quadratic sequence, will the first difference be
equals to 109
BY MR M. SHABALALA @NOMBUSO HIGH Page 3
, 1.4 Given the quadratic sequence : 𝑚 ; 5 ; 𝑛 ; 19; … ..
1.4.1 Calculate the value(s) of 𝑚 and 𝑛 if the second differece is 2.
1.4.2 Determine the 𝑛𝑡ℎ term of the quadratic sequence.
1.4.3 If 𝑇𝑛 = 𝑛2 + 𝑛 − 1 , determine the value of 121st term.
1.4.4 Determine the first term of the sequence that will have a value greater than 10301.
1.5 The following are the 4 terms of a quadratic sequence: 6 ; 5+𝑥; −6 ; and 6 𝑥
1.5.1 Show that 𝑥 = −3
1.5.2 Determine an expression of the general term of the sequence.
1.6 3 − 𝑡 ; −𝑡 ; √9 − 2𝑡 are the first three terms of an arithmetic sequence.
Determine the value of 𝑡.
1.7 Determine the next two terms in the following sequence: 1; 3 ; 9 ; 27; ____; _____
1.8 Determine the next two terms in the following sequence: 6; 12 ; 24 ; 48 ; 96 ____; _____
Can you see that: There is a common ratio (r) =2
𝑇1 = 6
𝑇2 = 6 × 21
𝑇3 = 6 × 2 × 2 = 6 × 22
𝑇5 = 6 × 24
𝑇10 = __________
𝑇100 = __________
𝑇𝑛 = __________
∴The above sequence is new in grade 12 and it is called Geometric Sequence
GEOMETRIC SEQUENCE
𝑻 𝑻 FORMULAR:
Common ratio (𝑻𝟐 = 𝑻𝟑 )
𝟏 𝟐 𝑇𝑛 = 𝑎𝑟 𝑛−1
1
1.9 Determine the value of 𝑏 and nth term of the following geometric sequence: − 4 ; 𝑏 ; −1; …
1.10 A geometric sequence has 𝑇3 = 20 and 𝑇4 = 40:
Determine: a) Common ratio b) A formula for 𝑇𝑛
BY MR M. SHABALALA @NOMBUSO HIGH Page 4
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