Sequences AND Series Practice Questions 2 PDF

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Bunch of questions for mathematics in practice for blooms taxonomy...

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MATHEMATICS 2021 REVISION MATERIAL

SEQUENCES AND SERIES

A collection of questions from previous question papers (2016 to 2020). Prepared by T Faya

Take note: Coding of questions KZN J16 – KZN June 2016 KZN S16 – KZN September 2016 ECS18 – Eastern Cape September 2016 NNM17 – National March 2017 And so on…………….

1

QUESTION 2 KZN J16 Given the quadratic sequence : 6 ; 6 ; 10 ; 18

; ....

2.1

Determine a formula for the nth 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) [12]

QUESTION 14 The following sequence of numbers forms a quadratic sequence. – 3 ; – 2 ; – 3 ; – 6 ; – 11 ; ……………….. 14.1. The first differences of the above sequence also form a sequence. Determine an expression for the general term of the first differences.

(3)

14.2. Calculate the first difference between the 35th and the 36th terms of the quadratic sequence.

(5)

14.3. Determine an expression for the nth term of the quadratic sequence.

(4)

14.4. Explain why the sequence of the numbers will never contain a positive term.

(4) [16]

QUESTION 2 FS J16 A pattern with a constant second difference has 𝑇𝑘 = 3𝑘 − 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.

2.2

Determine the 10𝑡ℎ term of this pattern.

(6) (2) [8]

QUESTION 3 KZN J16 3.1

If Sn  2n2  3n , calculate the 20th term of the series.

3.2

Given the series : 3 + 5 + 6 + 5 + 12 + 5 + …

(4)

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

QUESTION 3 FS J16 3.1

Given the arithmetic series: −14 − 11 − 8 + … 103. Calculate:

3.2 3.3

3.4

3.1.1

The 𝑘 𝑡ℎ term of this series.

(3)

3.1.2

How many terms are in this series?

(2)

Prove that: 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + ⋯ 𝑎𝑟 𝑛−1 + 𝑎𝑟 𝑛 =

𝑎(𝑟 𝑛−1) 𝑟−1

, where 𝑟 ≠ 1.

Calculate the sum of the areas of seven squares if the side of the first square is 36 cm, the side of the second square is 18 cm, and so on to the last one. Calculate the value 𝑝 if: ∞

(6)

9

𝑟

∑ 27𝑝 = ∑ (9 − 3𝑚)

𝑟=1

(4)

𝑚=3

(7) [22]

QUESTION 2 GPJ16 Given the sequence: 2.1 2.2

2.3

5; 12; 21; 32 .....

Determine the formula for the 𝑛𝑡ℎ term of the sequence. Determine between which two consecutive terms in the sequence the first difference will equal 245. Sketch a graph to represent the second differences.

(4)

(5) (2) [11]

QUESTION 3 GPJ16 3.1

Given: 16 + 8 + 4 + 2 +……. 3.1.1

Determine the sum of the first forty (40) terms of the series.

(3)

3.1.2

Write the given series in sigma notation.

(2)

3.1.3

Explain why the series converges.

(2)

350

3.2

Calculate:

 1  3k  k 3

200



 D 6x  

(6)

x

t 1

[13]

3

ECS16

QUESTION 2 FSS16 13

2.1

Calculate

3

(2)

r 4

2.2

2.3

Given the arithmetic sequence: 3; b; 19; 27; ... 2.2.1 Calculate the value of b.

(1)

2.2.2 Determine the n th term of the sequence.

(2)

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)

The above sequence 3; b; 19; 27; ... forms the first differences of a quadratic sequence. The first term of the quadratic sequence is 1. 2.3.1 Determine the fourth term( T4 ) of the quadratic sequence.

(2)

th 2.3.2 Determine the n term of the quadratic sequence.

(4)

2.3.3 Calculate the value of n if Tn  1  7700

4

(3) [18]

GPS16

QUESTION 2 LPS16 The 7th term of a geometric series is

1

128

and the 11th term is

1

2048

.

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 𝑇1 + 𝑇3 + 𝑇5 + ⋯ =

1 1 1    ... 2 8 32

from the above sequence. Calculate the sum to infinity of this new series.

(4) [10]

5

QUESTION 3 LPS16 A quadratic sequence is defined with the general term: 𝑛

𝑇𝑛 = ∑(6𝑘 − 2) . 𝑘=0

3.1

Show that 𝑇3 = 28

3.2

The sequence defined has the first 4 terms:

(3)

2 ; 12 ; 28 ; 50 Show that 𝑇𝑛 = 3𝑛 2 + 𝑛 − 2.

(4)

3.3

Calculate 𝑇40 of this sequence.

(2)

3.4

Calculate the first difference between 𝑇𝑛 and 𝑇𝑛−1 where 𝑇𝑛 = 442.

(5) [14]

MPS16

6

QUESTION 2 NWS16

Consider the following quadratic sequence: 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; ...

2.1.1

2.1.2

2.1.3

Write down the first 3 terms of the sequence of first differences of the quadratic sequence.

(1)

Write down the 100th term of the sequence of first differences of the quadratic sequence.

(1)

If x = 2, determine the general term of the quadratic sequence.

(4)

54 ; x ; 6 are the first three terms of a geometric sequence.

2.2.1

Calculate x.

(2)

2.2.2

Is this geometric sequence convergent? Motivate your answer by clearly showing all your calculations.

(3)

Determine the value of k for which: 60

 (3r

5

 4) 

r 5

k

(5)

p  2

Consider

4;

3 1 1 ;4; ;4; ; ... 4 4 12

which is a combination of 2 geometric patterns. 2.4.1 If the pattern continues in the same way, write down the next TWO terms in the sequence. 2.4.2

Calculate the sum of the first 25 terms of the sequence. Show all calculations.

(1)

(6) [23]

QUESTION 2 WCS16 2.1

Given: 3𝑥 − 1 2𝑥 − 1 7𝑥 − 5 ; ; 4 3 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)

7

2.2

Given the series: 18 + 6 + 2 + ⋯ 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, T10.

(2)

2.2.3

Determine 𝑆∞ − 𝑆10 .

(5 ) [15]

QUESTION 3 WCS16 3.1

Determine the value of: 33

∑(1 − 2𝑘)

(3)

𝑘=2

3.2

6; 5 + 𝑥; −6 and 6𝑥 form the first 4 terms of a quadratic sequence. 3.2.1

Show that 𝑥 = −3.

3.2.2

Determine an expression for the general term of the sequence.

(4) (4) [11]

QUESTION 2 NM16 2.1

2.2

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 nth term of this sequence.

(5)

2.1.3

Which term of the sequence will be equal to 322?

(4)

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.

8

(3) [18]

QUESTION 3 NM16 Chris bought a bonsai (miniature tree) at a nursery. When he bought the tree, its height was 130 mm. Thereafter the height of the tree increased, as shown below. INCREASE IN HEIGHT OF THE TREE PER YEAR

3.1

3.2

3.3

During the first year

During the second year

During the third year

100 mm

70 mm

49 mm

Chris noted that the sequence of height increases, namely 100 ; 70 ; 49 ..., was geometric. During which year will the height of the tree increase by approximately 11,76 mm?

(4)

Chris plots a graph to represent the height hn of the tree (in mm) n years after he bought it. Determine a formula for hn.

(3)

What height will the tree eventually reach?

NJ16

NJ16

9

(3) [10]

NN16

QUESTION 2 KZNJ17 The first difference sequence of the quadratic sequence is 3; 7; 11; . . . and the 51st term of the quadratic sequence is 5052. 2.1

Calculate the nth term of the quadratic sequence.

(6)

2.2

Which term of the quadratic sequence is equal to 20102?

(4) [10]

QUESTION 3 KZNJ17 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) [7]

QUESTION 4 KZNJ17 4.1

4.2

4.3

Prove that the sum to n terms of the arithmetic series whose first is “a” and its common difference is “d” is given by n Sn  2a  (n  1)d  2 3 In a series, S n  n2  4 n, calculate the value of the eighth term in the series. 2 Given the arithmetic sequence 4 ; 9 ; 14 ; 19 ; …. Determine the first term in the sequence that will be greater than 2017.

10

(4)

(4)

(3) [11]

QUESTION 2 KZNS17 2.1

Given below is the combination series of an arithmetic and a constant pattern: 2  3  5  3  8  3  ...

2.2

2.1.1

If the pattern continues, write down the next two terms.

(2)

2.1.2

Determine the 85th term of the given series.

(3)

2.1.3

Calculate the sum of the first 85 terms of the series.

(3)

Given the series ( x  2)  ( x 2  4)  ( x 3  2 x 2  4 x  8)  ... ( x  2 ). 2.2.1

Determine the values of 𝑥 for which the series converges.

2.2.2

Explain why the series will never converge to zero.

(4) (3) [15]

QUESTION 3 KZNS17 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 48th 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.

11

(2) [10]

NN17

12

KZNM18

QUESTION 2 KZNS18 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 nth term of the sequence.

(4)

2.3

Prove that all the terms of the quadratic sequence are odd.

(3) [9]

13

QUESTION 3 KZNS18

3  t ;  t ; 9  2t 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) [7]

QUESTION 4 KZNS18 4.1

4.2

Given the infinite geometric series x  3  x  3   x  3   ... 2

3

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)

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) [9]

ECS18

ECS18

14

QUESTION 2 FSS18

2.4

2.5

Consider the arithmetic sequence: 3 ;  2 ;  7 ;  12 ; ... 2.4.1 Calculate the 21st term of the sequence.

(2)

2.4.2 Which term of the sequence is equal to  177?

(2)

The sum of the first terms in an arithmetic series is given by: Sn  n 2  2n Calculate:

2.6

2.5.1 the sum of the first 13 terms.

(2)

2.5.2 the 13th term.

(2)

Given the quadratic pattern: x ; 6 ; 9 ; y ; 24 ; ... Calculate the sum of x and y.

(5) [13]

QUESTION 3 FSS18 3.1

n Prove that a  ar  ar 2  ... (to n terms) = a1  r  , r  1

3.2

Given:

1 r



(4)

 4 0,2

k 1

k 1

3.2.1 Write down the first THREE terms of the series.

(1)

3.2.2 Calculate the sum to infinity of the series.

(3)

3.2.3 Hence calculate the smallest number of terms of the series whose sum will differ by less than 0,0001 from the sum to infinity of the series.

(5) [13]

15

GPS18

LPS18

16

LPS18

MPS18

17

NCS18

NCS18

18

NWS18

NWS18

19

WCS18

WCS18

20

QUESTION 2 NM18 2.1

2.2

Given the following geometric sequence:

30 ; 10 ;

10 ;… 3

2.1.1

Determine n if the nth term of the sequence is equal to

2.1.2

Calculate: 30 10

10 . 729

10  3

(4)

(2)

Derive a formula for the sum of the first n terms of an arithmetic sequence if the first term of the sequence is a and the common difference is d.

(4) [10]

QUESTION 3 NM18 The first three terms of an arithmetic sequence are

– 1 ; 2 and 5.

3.1

Determine the nth term, Tn , of the sequence.

(2)

3.2

Calculate T43 .

(2)

3.3

Evaluate

n

T

k

in terms of n.

(3)

k 1

3.4

A quadratic sequence, with general term Tn , has the following properties:  T11  125  Tn  Tn 1  3n  4 (6) [13]

Determine the first term of the sequence.

QUESTION 2 NJ18 2.1

2.2

Given the quadratic pattern: 5 ; 10 ; 17 ; 26 ; … 2.1.1

Write down the next TWO terms of the pattern.

(2)

2.1.2

Determine the formula for the n th term of the pattern.

(4)

2.1.3

Which term of the pattern will have a value of 1 765?

(4)

The first 24 terms of an arithmetic series are: 35 + 42 + 49 + … + 196. Calculate the sum of ALL natural numbers from 35 to 196 that are NOT divisible by 7.

21

(5) [15]

QUESTION 3 NJ18 Themba is planning a bicycle trip from Cape Town to Pretoria. The total distance covered during the trip will be 1 500 km. He plans to travel 100 km on the first day. For every following day he plans to cover 94% of the distance he covered the previous day. 3.1

What distance will he cover on day 3 of the trip?

(2)

3.2

On what day of the trip will Themba pass the halfway point?

(4)

3.3

Themba must cover a certain percentage of the previous day's distance to ensure that he will eventually reach Pretoria. Calculate ALL possible value(s) of this percentage.

(3) [9]

QUESTION 2 NN18 2.1

2.2

Given the quadratic sequence: 2 ; 3 ; 10 ; 23 ; … 2.1.1

Write down the next term of the sequence.

(1)

2.1.2

Determine the nth term of the sequence.

(4)

2.1.3

Calculate the 20th term of the sequence.

(2)

Given the arithmetic sequence: 35 ; 28 ; 21 ; … Calculate which term of the sequence will have a value of –140.

2.3

For which value of n will the sum of the first n terms of the arithmetic sequence in QUESTION 2.2 be equal to the nth term of the quadratic sequence in QUESTION 2.1?

(3)

(6) [16]

QUESTION 3 NN18 A geometric series has a constant ratio of

1 and a sum to infinity of 6. 2

3.1

Calculate the first term of the series.

(2)

3.2

Calculate the 8th term of the series.

(2)

3.3

Given:

n

 32

1 k

 5,8125

Calculate the value of n.

(4)

k 1

20

3.4

If

 32  k 1

1 k

20

 p , write down

 242

k

in terms of p. (3) [11]

k 1

22

QUESTION 2 KNZJ19 The first four terms of the first difference of a quadratic sequence are 5 ; 9 ; 13 ; 17 ;... and the 61st term of the quadratic sequence is 7383.

2.1

Calculate the nth of the quadratic sequence.

(5)

2.2

Determine between which two consecutive terms is the 1st difference equal to 2021?

(3) [8]

Q...