WTS TUTORING 1
WTS TUTORING
NUMBER
PATTERNS
GRADE :12
COMPILED BY : PROF KWV KHANGELANI SIBIYA
CELL NO. : 0826727928
EMAIL : kwvsibiya@gmail.com
FACEBOOK P. : WTS MATHS & SCEINCE TUTORING
WTSWHATSAPP G. : 082 672 7928
WEBSITE : www.wtstutor.co.za
WTS TUTORING 2
NUMBER PATTERNS
Key words:
1. Sum : the result of addition
2. Difference : the result of subtraction
3. Product : the result of multiplication
4. Natural numbers : whole numbers greater than or equal to 1
5. Integer : a positive or negative whole number or zero.
6. Even numbers : any integer that can be divided exactly by 2
7. Odd numbers : any integer that cannot be divided exactly by 2
8. Factor : a number that divides exactly into a whole number with no
remainder
9. Divisible by : a number is divisible if, after diving, there is no remainder.
10. Prime number : numbers that have only two factors, the number itself and 1.
11. Multiple : the product of two natural numbers
12. Perfect squares; are rational numbers which can example end with 1;4;16 ;….but
never end in a 3 or an 8 etc
13. Non-Perfect squares; are irrational numbers which can example end with 3 or 8 but
never end in a 4 or an 16 etc
14. Which term: that is a number of terms
15. nth term: that is general term
16. determine the 10 th term; same as
17. existence of the sequence or series; the common ratio is between -1 and 1 or
18. greater than; the first term to be greater then : a if a is the given nth term
19. less than; the first term to be less then : a if a is the given nth term
20. never contain a positive or negative term; calculate the maximum value of the nth
term ( y value of turning point)
21. converging series; if and only if
WTS TUTORING 3
A. ARTHMETIC SEQUENCE
It is a sequence where the common different (d) between consecutive terms is constant.
GENERAL / N-TERM/ LAST TERM ( )
First common difference : (more useful if
variables are given / and if the unknown must be calculated)
General term: ( )
Given the last term: implies that is given and u can calculate the first term or
number of terms
If other terms are not given use the following: , ,
NB: hence solving simaltenouesly
: the value of the specific term
: number of the term in a sequence and is a counting number ( the position of n-th
term in the sequence)
SERIES
When we add the terms of a sequence together, we form a series. We use the symbol to
show the sum of the first n terms of a series. So
WTS TUTORING 4
Proof:
General formula : [ ( ) ] or [ ]
: the sum of terms
the last term
Kwv 1
Prove that: [ ( ) ]
Kwv 2
The sequence is given.
a. Determine the general term
b. Which term is equal to 71?
c. Which first term will be greater than 41?
d. Determine the sum of the first n terms
e. Hence , calculate the sum of the first 40 terms.
Kwv 3
How many terms of the series 3 + 8 + 13 + … must be added to give a sum of 2265?
WTS TUTORING 5
Kwv 4
The following arithmetic sequence is given: 20 ; 23 ; 26; 29 ;…………;101
a. How many terms are there in this sequence?
b. The even numbers are removed from the sequence. Calculate the sum of the terms of
the remaining sequence.
Kwv 5
Consider the following pattern:
1 2 3 6
45 6 15
7 89 24
10 11 12 33
Calculate the sum of the terms in the 2010th row.
Kwv 6
Consider an arithmetic sequence which has the second term equal to 8 and the T5 = 10
a. Determine the first term and common difference of this sequence
b. Determine the term.
c. Determine the sum of the first 50 terms
WTS TUTORING 6
Kwv 7
Given the finite arithmetic sequence: 5 ; 1 ; –3 ; ... ; –83 ; –87
a. Write down the fourth term (T4) of the sequence.
b. Calculate the number of terms in the sequence.
c. Calculate the sum of all the negative numbers in the sequence.
d. Consider the sequence: 5 ; 1 ; –3 ; ... ; –83 ; –87 ; … ; –4 187.
e. Determine the number of terms in this sequence that will be exactly divisible by 5
Kwv 8
The following is an arithmetic sequence:
a. Calculate the value of k
b. Write down the value of a and d
c. Explain why none of the numbers in this sequence are perfect squares.
Kwv 9
Determine the value(s) of x in the interval [ ] for which the sequence
-1 ; 2sin3x ; 5 ;..... will be arithmetic.
WTS TUTORING 7
B. GEOMETRIC SEQUENCE
It is a sequence whereby there is a common ratio (r) between consecutive terms.
GENERAL / N-TERM/ LAST TERM ( )
Common ratio: (more useful if variables are given / and if the unknown
must be calculated hence orlando pirates sign : kwv rule)
General term:
Given the last term :implies that is given and u can calculate the first term or
number of terms
If few terms are given: , , NB: solving simaltenouesly
Take notes of exponential and logarithms laws
SERIES
When we add the terms of a sequence together, we form a series. We use the symbol to
show the sum of the first n terms of a series. So
WTS TUTORING 8
Proof:
( )
For:
( )
For:
( ) ( )
General formula : or
Take note there is no different between two formulae
WTS TUTORING 9
Converges series
Is for geometric pattern
Note: { r is a common ratio }
Always work with inequalities
Sum to infinity:
Kwv 1
a(r n 1)
Prove that: a ar ar 2 ...(to n terms) ; r 1
r 1
Kwv 2
Given: ( ) ( ) ( ) as a geometric sequence and x 2
i) Determine the general term of the series in terms of x
ii. Calculate the value of x for which the sequence converges.
iii. Determine the sum to infinity of the series if x = 2,5.
Kwv 3
Given the geometric series:
i) If x = 4, then determine the sum to 15 terms of the sequence.
ii) Determine the values of x for which the original series converges.
iii) Determine the values of x for which the original series will be
increasing.
WTS TUTORING 10
Kwv 4
Given: 2 and –1, as the first two terms of an infinite geometric series. Calculate the sum of
this series.
Kwv 5
The following information of a geometric pattern is given
and
Determine the following:
i. numerical values of the first three term if r > 0
ii. n-term formula
Kwv 6
Mr KWV bought a bonsai (tree) at a nursery, when he bought the tree, its height was 130
mm, thereafter the height of the tree increased each year respectively:
100mm ; 70mm ; 49mm; …
i. During which year will the height of the tree increase by approximately 11,76mm?
ii. Mr KWV plots a graph to represent the height ( ) of the tree (in mm) in years
after he bought it. Determine a formula for ( )
iii. What height will the tree eventually reach?
WTS TUTORING 11
C. QUADRATIC PATTERN
It is a sequence whereby there is a constant second difference (d).
GENERAL TERM
Second common difference :
If the unknown given workout the second difference using given variable and hence
equate the second difference in order to solve the unknown.
General term: ( NB: substitute the values at the end)
Key
Given the implies that you can calculate the number of terms
If few terms are given substitute into: and hence solve
simaltenouesly
WTS TUTORING 12
Determine between two consecutive terms
To calculate the first difference between given terms
Using linear nth term
The term will always lies in the lowest number
Firstly calculate the n term and substitute with :
Using quadratic nth term
Simple substitute into the nth term by calculating the difference starting with lower
number.
To calculate two terms between
Firstly calculate the number of terms and substitute into nth term and take note to
increase the number for the difference.
NB:
Kwv 1
1. Given the quadratic sequence:
i. Write down the next TWO terms.
ii. Calculate the nth term of the quadratic sequence.
iii. Determine T10 of the above sequence.
iv. Which term in the sequence is equal to 55?
WTS TUTORING 13
v. Determine between which two consecutive terms of the quadratic sequence the
first difference will be equal to 2018.
vi. What is the value of the first term of the sequence that is greater than 77?
Kwv 2
The quadratic pattern is given. Determine the value of .
Kwv 3
The first four terms of a quadratic number pattern are
i. Calculate the value (s) of .
ii. If , determine the position of the first term in the quadratic number
pattern for which the sum of the first differences will be greater than 250.
Kwv 4
Dots are arranged to form a sequence of patterns as shown below:
Pattern 1 Pattern 2 Pattern 3 Pattern 4
i. If the pattern behaves consistently, write down the number of dots in pattern 5
ii. Determine a formula for the number of dots in the nth pattern.
iii. Use your formula to calculate which pattern number has 1 985 dots in it?
WTS TUTORING 14
Kwv 5
The following sequence of numbers forms a quadratic sequence:
-3 ; -2 ; -3 ; -6 ; -11 ;……………………………
a. The first differences of the above sequence also form a sequence. Determine an
expression for the general term of the first differences.
b. Calculate the first difference between the 35th and 36th terms of the quadratic pattern
c. Determine an expression for the nth term of the quadratic sequence.
d. Explain why the sequence will never contain a positive term.
Kwv 6
A quadratic sequence is defined with the following properties
a. Write down the value of:
1.
2.
b. Calculate the value of if
WTS TUTORING 15
Kwv 7
a. The nth term of a sequence is given by : ( )
1. Write down the first THREE terms of the sequence.
2. Which term of the sequence will have the greatest value?
3. What is the second difference of this quadratic sequence?
4. Determine ALL values of n for which the terms of the sequence will be less than -110.
D. COMBINATION OF AP & GP
Separate the sequence into two
Divide the number of terms in a counting number form and consider the odd and even
position.
The constant sequence will be given by :
The number of terms :
WTS TUTORING 16
Kwv 1
Given the combined arithmetic and constant sequence:
3 ; 2 ; 6 ; 2 ; 9 ; 2 ; ...
i. Write down the next TWO terms in the sequence.
ii. Calculate the sum of the first 100 terms of the sequence.
iii. Calculate the sum of the first 45 terms of the sequence.
iv. Determine the 85th term
Kwv 3
Consider the sequence: ;4; ; 7; ; 10 ;……
a. If the pattern continues in the same way, write down the next TWO terms in the sequence.
b. Calculate the sum of the first 50 terms of the sequence
Kwv 3
Given the sequence:
Determine the value(s) of if the sequence is:
i. Arithmetic
ii. Geometric
WTS TUTORING 17
Kwv 4
The following is an arithmetic sequence:
a. Calculate the value of p
b. Write down the value of:
1. The first term of the sequence.
2. The common difference.
c. Explain why none of the numbers in this arithmetic sequence are perfect squares.
Kwv 5
Given the geometric series: 256 + p + 64 – 32 + ……….
a. Determine the value of p
b. Calculate the sum of the first 8 terms of the series.
c. Why does the sum to infinity for this series exist.
d. Calculate
Kwv 6
Log 2 and log 4 are the first 2 terms of arithmetic as well as a geometric sequence. Determine
an expression for the nth term of each sequence. Simplify each answer to one term
E.GIVEN SUM FORMULA
Take note of the formula of calculating the term :
Be in position to calculate the first three terms from the given sum formula, hence
general term
WTS TUTORING 18
Kwv 1
The sum to n terms of a sequence of numbers is given as: S n
n
5n 9
2
i. Calculate the first 3 terms
ii. Determine the nth tern
iii. Calculate the sum to 23 terms of the sequence.
iv. Hence, calculate the 23rd term of the sequence.
Kwv 2
The sum of the first n terms of a series is given by the formula
a. Determine the sum of the first 6 terms
b. Determine the first 3 terms of the sequence.
F.INTERPRETATIONS
Key:
Given the last term: implies that is given and you can calculate the first term or
number of terms
If other terms are omitted use the following key : , ,
and or : , ,
NB: hence solving simaltenouesly
WTS TUTORING 19
Kwv 1
The first two terms of a geometric sequence and an arithmetic sequence are the same. The
first term is 12. The sum of the first three terms of the geometric sequence is 3 more than the
sum of the first three terms of the arithmetic sequence. Determine TWO possible values for
the common ratio, r, of the geometric sequence
Kwv 2
8
The first two terms of an infinite geometric sequence are 8 and . Prove, without the use
2
of a calculator, that the sum of the series to infinity is 16 8 2 .
Kwv 3
Three numbers form a geometric sequence. Their sum is 21 and their product is 64.
Find the numbers (show all working)
Kwv 4
The first two terms of a geometric sequence are the same as the first two terms of an
arithmetic sequence. The first term is 8 and is greater than the second term. The sum of the
first three terms of the arithmetic sequence is 1,125 less than the sum of the first three terms
of the geometric sequence. Determine the first three terms of each sequence.
WTS TUTORING 20
G.SIGMA NOTATION
Means the sum of n-th terms
: stand for last value after substituting in the n-th terms
: stand for the first value after substituting in the n-th terms
: stand for the general term
NB: always determine the first 3 terms in order to know the type of the pattern e.g. d-
arithmetic or r-geometric
Number of terms :
Given the sum: you can calculate the number of terms
If the question says evaluate or calculate it means determine the sum of n-th terms
For the sum: First three terms-hence the pattern-no. of terms –the formula
Writing in sigma notation: First three terms-hence the pattern-no. of terms –and lastly
for the first value
Kwv 1
. The following geometric series is given: up to 5 terms.
i. Write down the series in sigma notation.
ii. Calculate the sum to 5 terms of the series.
WTS TUTORING 21
Kwv 2
Given: ( ) ( )
i. For what values of will the series converge?
ii. Hence, determine ∑ ( ) if
Kwv 3
n
Calculate the value of n if: 23
k 1
k 1
531440
Kwv 4
The following geometric series is given: x = 5 + 15 + 45 + … to 20 terms.
i. Write the series in sigma notation.
ii calculate the value of x
Kwv 5
Evaluate:
∑( )
Kwv 6
For which values of will :
∑
WTS TUTORING 22
Kwv 7
Determine the value of if:
∑ ∑( )
EXAM GUIDLINES
NUMBER PATTERNS, SEQUENCES AND SERIES
1. The sequence of first differences of a quadratic number pattern is linear. Therefore,
knowledge of linear patterns can be tested in the context of quadratic number patterns.
2. Recursive patterns will not be examined explicitly.
3. Links must be clearly established between patterns done in earlier grades.
WTS TUTORING 23
YOUR NOTES:
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WTS TUTORING 24
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WTS TUTORING 25
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