Class 9 Maths Exploring Some More Progressions Notes

Class 9 Maths Exploring Some More Progressions Notes

Chapter 6 — Exploring Some More Progressions

This chapter extends the study of sequences and progressions beyond the basic ideas already learned.

1. Progression as an Ordered Pattern

A progression is a sequence whose terms follow a definite mathematical pattern.

The important task is to identify how one term is generated from another.

2. Arithmetic Progression

An arithmetic progression (AP) has a constant difference between consecutive terms.

If the first term is a and common difference is d:

a, a+d, a+2d, a+3d, ...

The nth term is:

aₙ = a + (n−1)d

The sum of the first n terms is:

Sₙ = n/2 [2a + (n−1)d]

An alternative form is:

Sₙ = n/2 (a + l)

where l is the last term.

3. Geometric Progression

A geometric progression (GP) has a constant ratio between consecutive terms.

If first term = a and common ratio = r:

a, ar, ar², ar³, ...

The nth term is:

aₙ = arⁿ⁻¹

For a finite GP:

Sₙ = a(rⁿ−1)/(r−1), when r ≠ 1

Equivalent forms may be used depending on whether r is greater or less than 1.

4. Harmonic Progression

A sequence is in harmonic progression when its reciprocals form an arithmetic progression.

So the useful strategy is:

To analyse an HP, examine the reciprocals of its terms.

5. Recognising a Progression

When given a sequence, don’t immediately apply a formula.

First determine:

  1. Is the difference constant?
  2. Is the ratio constant?
  3. Do the reciprocals form an AP?
  4. Is there another stated pattern?

Only after identifying the pattern should you choose the appropriate formula.

Exam Focus

The central skill in progression questions is pattern recognition, followed by correct selection of the nth-term or sum formula.

Chapter 6 — Exploring Some More Progressions: Complete Question Bank

A. Multiple Choice Questions (MCQs)

1. Which of the following is a geometric progression?

A. 2, 4, 6, 8
B. 3, 6, 12, 24
C. 1, 4, 9, 16
D. 5, 7, 10, 14

2. In a GP, the ratio between two consecutive terms is called the:

A. Common difference
B. Common ratio
C. Common sum
D. Common factor

3. The general form of a GP is:

A. a, a+d, a+2d,...
B. a, ar, ar², ar³,...
C. a, a², a³,...
D. a, r, a+r,...

4. The common ratio of 2, 6, 18, 54,... is:

A. 2
B. 3
C. 6
D. 9

5. The nth term of a GP is:

A. a+(n−1)d
B. arⁿ
C. arⁿ⁻¹
D. a+n r

6. The common ratio of 81, 27, 9, 3,... is:

A. 3
B. −3
C. 1/3
D. −1/3

7. If the first term of a GP is 5 and the common ratio is 2, the third term is:

A. 10
B. 15
C. 20
D. 25

8. If a=3 and r=4, then the fourth term is:

A. 48
B. 64
C. 192
D. 256

9. The sum of the first n terms of a GP with r≠1 is:

A. a+(n−1)r
B. a(rⁿ−1)/(r−1)
C. n(a+r)
D. arⁿ

10. If r=1, every term of the GP is:

A. Zero
B. Equal to the first term
C. Increasing
D. Decreasing

11. A GP with common ratio between 0 and 1 has terms that:

A. Increase indefinitely in magnitude
B. Remain constant
C. Decrease in magnitude
D. Alternate necessarily

12. For an infinite GP to have a finite sum, the absolute value of its common ratio must be:

A. Greater than 1
B. Equal to 1
C. Less than 1
D. Equal to 2

13. The sum to infinity of a + ar + ar² + ..., when |r|<1, is:

A. a/(1+r)
B. a/(1−r)
C. a(1−r)
D. ar/(1−r)

14. The sequence 4, −2, 1, −1/2,... has common ratio:

A. 2
B. −2
C. 1/2
D. −1/2

15. If |r|>1, the terms of a non-zero infinite GP generally:

A. Approach zero
B. Become unbounded in magnitude
C. Become equal
D. Always alternate between 0 and 1

B. Fill in the Blanks

16. A sequence in which the ratio of consecutive terms is constant is called a __________ progression.

17. The first term of a GP is generally denoted by __________.

18. The common ratio of a GP is generally denoted by __________.

19. The nth term of a GP is __________.

20. In a GP, t₂/t₁ = __________.

21. The first term of the sequence 7, 14, 28, 56,... is __________.

22. The common ratio of 5, 15, 45,... is __________.

23. If r=1, all terms of the GP are __________.

24. The sum of the first n terms of a GP depends on the first term, number of terms and the __________.

25. An infinite GP can have a finite sum when |r| is __________ than 1.

26. The sum to infinity of a convergent GP is __________.

27. A GP with a negative common ratio has terms whose signs generally __________.

28. In a, ar, ar²,..., the exponent of r in the nth term is __________.

C. True or False

29. Every arithmetic progression is also a geometric progression.

30. Every geometric progression has a constant common ratio.

31. 2, 4, 8, 16,... is a GP.

32. 3, 6, 9, 12,... is a GP.

33. The common ratio can be zero in a GP after the first term.

34. The nth term of a GP is arⁿ⁻¹.

35. A GP with common ratio 1 has all its terms equal.

36. The sequence 1, −1, 1, −1,... has common ratio −1.

37. Every infinite GP has a finite sum.

38. An infinite GP with |r|<1 has a finite sum.

39. If r=2, the terms of a non-zero GP approach zero.

40. In a GP, each term after the first can be obtained by multiplying the previous term by r.

D. Match the Following

Column AColumn B
41. First terma. arⁿ⁻¹
42. Common ratiob. a
43. nth termc. r
44. Finite GP sumd. a(1−rⁿ)/(1−r)
45. Infinite GP sume. a/(1−r)

E. Identify the GP

Determine whether each sequence is a GP. If it is, find its common ratio.

46. 3, 9, 27, 81,...

47. 5, 10, 20, 40,...

48. 16, 8, 4, 2,...

49. 2, 5, 8, 11,...

50. −3, 6, −12, 24,...

51. 1/2, 1/4, 1/8, 1/16,...

52. 7, 14, 21, 28,...

F. Find the nth Term

53. Find the 8th term of 2, 6, 18, 54,....

54. Find the 7th term of 5, 10, 20, 40,....

55. Find the 6th term of 81, 27, 9, 3,....

56. Find the 10th term of a GP with a=3 and r=2.

57. Find the 5th term of a GP with a=64 and r=1/2.

58. Find the 6th term of a GP with a=−2 and r=3.

59. If the first term is 7 and the common ratio is −2, find the fourth term.

60. If the third term of a GP is 20 and the first term is 5, find the possible positive common ratio.

G. Find the Missing Terms

61. Find the missing term:

3, 6, __, 24, 48

62. Find the missing term:

64, 32, __, 8, 4

63. Find the missing term:

2, __, 18, 54

64. Find the common ratio if:

4, __, 36, 108

65. Find x if 2, x, 18 are consecutive terms of a GP.

66. Find x if x, 12, 36 are consecutive terms of a GP and x is positive.

H. Sum of the First n Terms

67. Find the sum of the first 5 terms of:

2, 4, 8, 16,...

68. Find the sum of the first 6 terms of:

3, 6, 12, 24,...

69. Find the sum of the first 4 terms of:

5, 15, 45,...

70. Find the sum of the first 5 terms of:

81, 27, 9, 3,...

71. Find the sum of the first 6 terms when a=2 and r=3.

72. Find the sum of the first 5 terms when a=16 and r=1/2.

73. Find the sum of the first n terms of a GP with first term a and common ratio r, where r≠1.

74. What happens to the formula for the sum when r=1?

I. Infinite Geometric Progressions

75. Determine whether the following infinite GP has a finite sum:

4 + 2 + 1 + 1/2 + ...

76. Find the sum to infinity:

6 + 3 + 3/2 + 3/4 + ...

77. Find the sum to infinity:

10 + 2 + 0.4 + 0.08 + ...

78. Find the sum to infinity of a GP with a=12 and r=1/3.

79. Determine whether the GP with a=5 and r=−1/2 has a finite sum.

80. Find the sum to infinity of:

8 − 4 + 2 − 1 + ...

81. Explain why an infinite GP with r=2 does not have a finite sum.

82. Explain why an infinite GP with r=1/2 has a finite sum.

J. Application-Based Questions

83. A square has side 40 cm. A sequence of smaller squares is formed so that each side is multiplied by 1/√2. Explain why the side lengths form a GP.

84. A quantity starts at 100 and becomes half its previous value at every stage. Write the first five terms and identify the common ratio.

85. A ball is dropped from a height of 20 m and rebounds to half the previous height each time. Write the rebound heights as a GP.

86. A machine produces twice as many units each successive hour, beginning with 10 units. How many units are produced during the first 6 hours in total?

87. A savings amount grows by a fixed multiplicative factor each year. Explain why a GP may be an appropriate mathematical model.

88. A sequence of areas is 1600, 800, 400,.... Identify the first term and common ratio.

89. A model consists of layers whose areas follow 100, 50, 25, 12.5,.... Find the total area of all layers if the pattern continues indefinitely.

K. Conceptual Questions

90. What is the difference between an arithmetic progression and a geometric progression?

91. How can you test whether a given sequence is a GP?

92. Why must the common ratio be constant in a GP?

93. Explain the meaning of the expression arⁿ⁻¹.

94. Why does a negative common ratio produce alternating signs?

95. What is the difference between a finite GP and an infinite GP?

96. Why is the condition |r|<1 important for the sum to infinity?

97. Explain why the terms of a convergent infinite GP become progressively smaller in magnitude.

L. Assertion–Reason Questions

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

98. Assertion: 3, 6, 12, 24,... is a GP.
Reason: Each term is obtained by multiplying the previous term by 2.

99. Assertion: The sequence 2, 5, 8, 11,... is a GP.
Reason: The differences between consecutive terms are constant.

100. Assertion: The nth term of a GP is arⁿ⁻¹.
Reason: The first term contains r⁰, the second contains r¹, and so on.

101. Assertion: An infinite GP with |r|<1 has a finite sum.
Reason: Its terms approach zero as the number of terms increases.

102. Assertion: The infinite GP 1+2+4+8+... has a finite sum.
Reason: Its common ratio is greater than 1.

M. Higher-Order Thinking Questions

103. Three consecutive terms of a GP are x, 12, 48. Find x.

104. Three consecutive positive terms of a GP have product 216. If the middle term is 6, find the other two terms.

105. If the 3rd term of a GP is 12 and the 6th term is 96, find the common ratio.

106. The first term of a GP is 4 and its fourth term is 108. Find the positive common ratio.

107. A GP has first term 5 and common ratio 3. Which term is equal to 405?

108. The sum of the first three terms of a GP is 21 and the first term is 3. Find the possible positive common ratio.

109. An infinite GP has first term 9 and sum to infinity 12. Find its common ratio.

110. If an infinite GP has first term 20 and common ratio −1/4, find its sum to infinity.

111. Explain why the infinite GP 5−5+5−5+... cannot be assigned a finite sum using the ordinary GP sum-to-infinity formula.

112. A geometric sequence has a negative common ratio whose magnitude is less than 1. Describe the behaviour of its terms.

N. Mixed Challenge Questions

113. Find the sum:

1 + 3 + 9 + 27 + 81.

114. Find the sum:

64 + 32 + 16 + 8 + 4.

115. Find the sum of the first 8 terms of a GP with a=1 and r=2.

116. Find the sum to infinity:

12 + 6 + 3 + 3/2 + ...

117. Find the 9th term of a GP whose first term is 2 and common ratio is −3.

118. A GP has terms x, 2x, 4x,.... Find the sum of its first 7 terms in terms of x.

119. A sequence has first term 100 and common ratio 0.1. Find its sum to infinity.

120. Compare the behaviour of the infinite GPs with common ratios 1/2, −1/2, 2, and −2.

Answer Key

MCQs

  1. B
  2. B
  3. B
  4. B
  5. C
  6. C
  7. C
  8. C
  9. B
  10. B
  11. C
  12. C
  13. B
  14. D
  15. B

Fill in the Blanks

  1. geometric
  2. a
  3. r
  4. arⁿ⁻¹
  5. r
  6. 7
  7. 3
  8. equal
  9. common ratio
  10. less
  11. a/(1−r)
  12. alternate
  13. n−1

True/False

  1. False
  2. True
  3. True
  4. False
  5. True
  6. True
  7. True
  8. True
  9. False
  10. True
  11. False
  12. True

Match

41–b
42–c
43–a
44–d
45–e

GP Identification

  1. Yes, r=3
  2. Yes, r=2
  3. Yes, r=1/2
  4. No
  5. Yes, r=−2
  6. Yes, r=1/2
  7. No

Numerical Answers

  1. 4374
  2. 320
  3. 1/3
  4. 1536
  5. 4
  6. −486
  7. −56
  8. 2
  9. 12
  10. 16
  11. 6
  12. 3
  13. 6
  14. 4
  15. 62
  16. 189
  17. 605
  18. 121
  19. 728
  20. 31
  21. Sₙ = a(1−rⁿ)/(1−r)
  22. Sₙ=na
  23. Yes
  24. 12
  25. 12.5
  26. 18
  27. Yes
  28. 16/3
  29. Because the terms do not approach zero; their magnitudes increase.
  30. Because the terms approach zero and the infinite sum converges.

Application Answers

  1. 100, 50, 25, 12.5, 6.25; r=1/2
  2. 630 units
  3. First term = 1600; common ratio = 1/2
  4. 200

Higher-Order Answers

  1. 3
  2. 3 and 12
  3. r=2
  4. r=3
  5. 5th term
  6. r=2
  7. r=1/4
  8. 16
  9. The signs alternate while the magnitudes decrease toward zero.

Mixed Challenge Answers

  1. 121
  2. 124
  3. 255
  4. 24
  5. 13122
  6. 127x
  7. 1000/9
  • r=1/2: converges to a finite positive sum.
  • r=−1/2: converges with alternating signs.
  • r=2: diverges.
  • r=−2: diverges in magnitude with alternating signs.

Quick Revision Formula Sheet

Geometric Progression

a, ar, ar², ar³,...

nth Term

Tₙ = arⁿ⁻¹

Sum of First n Terms

For r≠1:

Sₙ = a(1−rⁿ)/(1−r)

If r = 1

Sₙ = na

Sum to Infinity

For |r|<1:

S∞ = a/(1−r)