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This document covers the revision material of Grade 12 past papers from 2016 to 2020 for Mathematics
Typology: Study Guides, Projects, Research
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Given the quadratic sequence : 6 ; 6 ; 10 ; 18 ;....
2.1 Determine a formula for the n
th term of the sequence. (4)
2.2 Determine between which two consecutive terms the first difference is 200? (4)
2.3 Which term in the quadratic sequence has a value of 32010? (4)
The following sequence of numbers forms a quadratic sequence.
14.1. The first differences of the above sequence also form a sequence.
Determine an expression for the general term of the first differences.
14.2. Calculate the first difference between the 35
th and the 36
th terms of the quadratic
sequence.
14.3. Determine an expression for the n
th term of the quadratic sequence. (^) (4)
14.4. Explain why the sequence of the numbers will never contain a positive term. (4)
A pattern with a constant second difference has 𝑇𝑘 = 3 𝑘 − 2 as the general term of its first
difference. The first term of the quadratic pattern is 7.
2.1 Determine the general term of the pattern. (6)
2.2 (^) Determine the 10 𝑡ℎ^ term of this pattern. (2)
3.1 (^) If S n n n^2
2 , calculate the 20
th term of the series. (4)
3.2 Given the series : 3 + 5 + 6 + 5 + 12 + 5 + …
3.2.1 Calculate the sum of the first 20 terms. (4)
3.2.2 Which term of the series is equal to 6291456? (4)
2.1 Calculate
13
4
r
2.2.1 Calculate the value of (^) b. (1)
2.2.2 Determine the
th
2.2.3 Calculate the value of the thirtieth term( T 30 ) (2)
2.2.4 Calculate the sum of the first 30 terms of the sequence. (2)
first term of the quadratic sequence is 1.
2.3.2 Determine the
th
2.3.3 Calculate the value of n if Tn ^1 ^7700 (3)
The 7
th term of a geometric series is
and the 11
th term is
If 𝑟 < 0 ,
2.1 Determine the first term of the sequence. (4)
2.2 Will this sequence converge? Explain. (2)
2.3 A new series is formed by taking 𝑇 1 + 𝑇 3 + 𝑇 5 + ⋯ = ...
32
from the above sequence.
Calculate the sum to infinity of this new series. (4)
x ; x 2 x ; x 2 x 3 ; x x 2 x 3 x 4 x ; x 2 x 3 x 4 x 5 ; ... x
60 5
5 2
r p
r k
^ (5)
2.1 Given:
3 𝑥 − 1
4
;
2 𝑥 − 1
3
;
7 𝑥 − 5
12
2.1.1 If 𝑥 = 5 , determine the values of the first three terms. (1)
2.1.2 What type of sequence is this? Give a reason for your answer. (2)
2.1.3 Which term will be equal to − 44 , 5? (3)
2.2 Given the series:
2.2.1 What is the value of the first negative term, if any? Explain your answer. (2)
2.2.2 Determine the tenth term, T 10. (2)
2.2.3 Determine 𝑆∞ − 𝑆 10. (5 )
[15]
QUESTION 3 WCS
3.1 Determine the value of:
∑( 1 − 2 𝑘)
33
3.2 6 ; 5 + 𝑥; − 6 and 6 𝑥 form the first 4 terms of a quadratic sequence.
3.2.1 (^) Show that 𝑥 = − 3. (4)
3.2.2 Determine an expression for the general term of the sequence. (4)
[11]
2.1 Given the following quadratic sequence: −2 ; 0 ; 3 ; 7 ; ...
2.1.1 Write down the value of the next term of this sequence. (1)
2.1.2 Determine an expression for the n
th term of this sequence. (5)
2.1.3 Which term of the sequence will be equal to 322? (4)
2.2 Consider an arithmetic sequence which has the second term equal to 8 and the fifth
term equal to 10.
2.2.1 Determine the common difference of this sequence. (3)
2.2.2 Write down the sum of the first 50 terms of this sequence, using sigma
notation. (2)
2.2.3 Determine the sum of the first 50 terms of this sequence. (3)
[18]
The first difference sequence of the quadratic sequence is 3; 7; 11;... and
the 51
st term of the quadratic sequence is 5052.
2.1 Calculate the n
th term of the quadratic sequence. (6)
2.2 Which term of the quadratic sequence is equal to 20102? (4)
3.1 Given the geometric sequence:
2 6 18 ... to 50 terms
3.1.1. Write down the next TWO terms in the sequence. (2)
3.1.2 Write down the series in sigma notation. (3)
3.1.3 Calculate the sum of the first 50 terms of the sequence. (2)
4.1 Prove that the sum to n terms of the arithmetic series whose first is “ a ” and its
common difference is “ d ” is given by
a n d
n Sn 2 ( 1 ) 2
In a series, Sn n 4 n 2
, calculate the value of the eighth term in the series.
4.3 Given the arithmetic sequence
4 ; 9 ; 14 ; 19 ; ….
Determine the first term in the sequence that will be greater than 2017. (3)
2.1 Given below is the combination series of an arithmetic and a constant pattern:
2.1.1 If the pattern continues, write down the next two terms. (2)
2.1.2 Determine the 85
th term of the given series. (3)
2.1.3 Calculate the sum of the first 85 terms of the series. (3)
2.2 Given the series( 2 ) ( 4 ) ( 2 4 8 ) ...
2 3 2 x x x x x ( x 2 ).
2.2.1 Determine the values of 𝑥 for which the series converges. (4)
2.2.2 Explain why the series will never converge to zero. (3)
[15]
QUESTION 3 KZNS
Given the quadratic sequence: 3; 5; 11; 21; x
3.1 Write down the value of x. (1)
3.2 Determine the value of the 48
th term. (5)
3.3 Prove that the terms of this sequence will never consist of even numbers. (2)
3.4 If all the terms of this sequence are increased by 100, write down the general term of
the new sequence. (2)
[10]
The first four terms of a quadratic sequence are^9 ;^19 ;^33 ;^51 ;...
2.1 Write down the next TWO terms of the quadratic sequence. (2)
2.2 Determine the n
th term of the sequence. (4)
2.3 Prove that all the terms of the quadratic sequence are odd. (3)
3 t ; t ; 9 2 t^ are the first three terms of an arithmetic sequence.
3.1 Determine the value of t. (4)
3.2 If t 8 , then determine the number of terms in the sequence that will be positive. (3)
4.1 Given the infinite geometric series 3 3 3 ...
2 3 x x x
4.1.1 Write down the value of the common ratio in terms of x. (1)
4.1.2 For which value(s) of x will the series converge? (3)
4.2 An arithmetic sequence and a geometric sequence have their first term as 3.
The common difference of the arithmetic sequence is p and the common ratio of the
geometric sequence is p. If the tenth term of the arithmetic sequence is equal to the
sum to infinity of the geometric sequence, determine the value of p. (5)
