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Additional Mathematics questions, chapter 5
Nama / NameKelas / ClassTarikh / Date

Progressions

43 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

14 marks
Choose the correct answer. Each question carries 1 mark.
1

Question 1

Which sequence is an arithmetic progression?
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B
Only 2, 5, 8, 11 has a constant difference (+3). 3, 6, 12, 24 and 1, ½, ¼, ⅛ have a constant ratio, so they are geometric.
For the teacher: Remembering, low difficulty, standard 5.1.1
Textbook p. 128
2

Question 2

The AP 7, 3, −1, −5, … has common difference d equal to
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A
d = T₂ − T₁ = 3 − 7 = −4. The terms are decreasing, so d is negative. 4 forgets the sign.
For the teacher: Understanding, low difficulty, standard 5.1.1
Textbook p. 129
3

Question 3

Find the 15th term of the arithmetic progression −4, 2, 8, …
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C
a = −4, d = 6. T₁₅ = −4 + 14(6) = 80. 86 uses 15d instead of 14d; 26 is T₆.
For the teacher: Applying, low difficulty, standard 5.1.2
Textbook p. 131
4

Question 4

How many terms are there in the arithmetic progression 5, 9, 13, …, 101?
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D
101 = 5 + (n − 1)(4), so n − 1 = 24 and n = 25. 24 forgets to add 1 back.
For the teacher: Applying, low difficulty, standard 5.1.2
Textbook p. 131
5

Question 5

Find the sum of the first 20 terms of the arithmetic progression 3, 7, 11, …
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A
S₂₀ = 20/2 [2(3) + 19(4)] = 10 × 82 = 820. 1 640 forgets to divide by 2.
For the teacher: Applying, low difficulty, standard 5.1.3
Textbook p. 133
6

Question 6

The sum of the first n terms of a progression is Sₙ = 2n² − 5n. Find the 9th term.
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B
T₉ = S₉ − S₈ = (162 − 45) − (128 − 40) = 117 − 88 = 29. 117 is S₉ itself.
For the teacher: Analysing, high difficulty, standard 5.1.3
Textbook p. 134
7

Question 7

3x − 1, 4x and 6x − 2 are three consecutive terms of an arithmetic progression. Find x.
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C
In an AP, 2(middle) = sum of the outer terms: 8x = 9x − 3, so x = 3. The terms are 8, 12, 16.
For the teacher: Applying, medium difficulty, standard 5.1.2
Textbook p. 131
8

Question 8

Which of these is a geometric progression?
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D
Each term is the previous one multiplied by ⅓. The others have a constant difference, so they are arithmetic.
For the teacher: Remembering, low difficulty, standard 5.2.1
Textbook p. 140
9

Question 9

Find the 6th term of the geometric progression 2, 6, 18, …
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A
r = 3. T₆ = 2 × 3⁵ = 2 × 243 = 486. 1 458 uses 3⁶.
For the teacher: Applying, low difficulty, standard 5.2.2
Textbook p. 142
10

Question 10

Find the sum of the first 6 terms of the geometric progression 2, 6, 18, …
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B
S₆ = 2(3⁶ − 1)/(3 − 1) = 728. 1 456 forgets to divide by r − 1 (it is 2 × 728); 486 is T₆ only.
For the teacher: Applying, low difficulty, standard 5.2.3
Textbook p. 144
11

Question 11

Find the sum to infinity of the geometric progression 81, 27, 9, …
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C
r = ⅓ so S∞ = 81 ÷ (1 − ⅓) = 81 ÷ ⅔ = 121.5. 243 is 81 ÷ ⅓: it uses r instead of 1 − r.
For the teacher: Applying, low difficulty, standard 5.2.4
Textbook p. 146
12

Question 12

The recurring decimal 0.777… written as a fraction in its simplest form is
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D
0.777… = 0.7 + 0.07 + 0.007 + … is a GP with a = 0.7, r = 0.1, so S∞ = 0.7 ÷ 0.9 = 7/9.
For the teacher: Analysing, medium difficulty, standard 5.2.4
Textbook p. 147
13

Question 13

The 3rd term of a geometric progression is 12 and the 5th term is 48. If all terms are positive, the first term is
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A
T₅ ÷ T₃ = r² = 4, r = 2 (positive). a = 12 ÷ r² = 12 ÷ 4 = 3.
For the teacher: Analysing, medium difficulty, standard 5.2.2
Textbook p. 142
14

Question 14

Which geometric progression has a sum to infinity?
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B
S∞ exists only when |r| < 1. 12, 6, 3 has r = ½. The others have r = 2, −2 and 1.
For the teacher: Understanding, low difficulty, standard 5.2.4
Textbook p. 146

Section B: structured questions

29 marks
Answer all questions. Show your working.
15

Question 15

The 3rd term of an arithmetic progression is 11 and the 8th term is 26.
[2 marks] (a)Find the common difference.
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Tick each point you got.
[1 mark] (b)Find the first term.
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Tick each point you got.
[2 marks] (c)Which term is equal to 146?
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Tick each point you got.
For the teacher: Applying, low difficulty, standard 5.1.2
Textbook p. 131
16

Question 16

Given the arithmetic series 4 + 7 + 10 + …
[2 marks] (a)Find the sum of the first 35 terms.
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Tick each point you got.
[2 marks] (b)Show that the sum of the first n terms is Sₙ = n(3n + 5)/2.
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Tick each point you got.
[3 marks] (c)Find the least number of terms for the sum to be more than 1 000.
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Check: S₂₆ = 13 × 83 = 1 079.
Tick each point you got.
For the teacher: Applying, medium difficulty, standard 5.1.3
Textbook p. 134
17

Question 17

Hana has two job offers for the next 2 years. Company A: RM2 800 in the first month, with an increase of RM30 every month. Company B: RM33 000 in the first year, with an increase of RM1 200 each year.
[2 marks] (a)Calculate Hana's total income from Company A in 24 months.
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Tick each point you got.
[1 mark] (b)Calculate her total income from Company B in 2 years.
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[1 mark] (c)Which company pays more over the 2 years, and by how much?
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For the teacher: Evaluating, low difficulty, standard 5.1.4
Textbook p. 138
18

Question 18

A ball is dropped and bounces. The first rebound reaches a height of 5 m, and each rebound is 60% of the one before it.
[2 marks] (a)Find the height of the 3rd rebound.
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Tick each point you got.
[3 marks] (b)From which rebound onwards is the height less than 0.5 m?
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Check: 5(0.6)⁴ = 0.648 m and 5(0.6)⁵ = 0.389 m.
Tick each point you got.
[2 marks] (c)The heights of all the rebounds form a geometric progression. Find the sum of the heights of all the rebounds if the ball bounced for ever.
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Tick each point you got.
For the teacher: Applying, medium difficulty, standard 5.2.4
Textbook p. 148
19

Question 19

A geometric progression has first term 12 and sum to infinity 18.
[2 marks] (a)Find the common ratio.
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Tick each point you got.
[1 mark] (b)Find the 3rd term.
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[3 marks] (c)Express 0.121212… as a fraction in its simplest form using the sum to infinity of a geometric progression.
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Tick each point you got.
For the teacher: Analysing, medium difficulty, standard 5.2.4
Textbook p. 146