Sequence & Series

 

Sequence & Series: Comprehensive Notes and 100 Practice Questions with Answers

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1. What are Sequences and Series?

• Sequence: An ordered list of numbers following a specific pattern or rule. Each number is called a term. Sequences can be finite or infinite. Example: 2, 4, 6, 8, ... (even numbers)1235.

• Series: The sum of the terms of a sequence. Example: 2 + 4 + 6 + 8 + ... is a series1235.

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2. Types of Sequences

A. Number Series (Missing/Wrong Terms)

• Number series are sequences where each term follows a rule. You may be asked to find the next term, a missing term, or a wrong term.

  • Example (Missing): 2, 4, 8, 16, ? (Answer: 32, as each term ×2)

  • Example (Wrong): 3, 6, 12, 25, 48 (Here, 25 is wrong; should be 24)

B. Arithmetic Progression (AP)

• Definition: Each term after the first is obtained by adding a fixed number (common difference, d) to the previous term12367.

• General term: $a_n=a_1+(n-1)d$

• Sum of first n terms: $S_n=\frac{n}{2}[2a_1+(n-1)d]$

• Example: 3, 7, 11, 15, ... (d = 4)

C. Geometric Progression (GP)

• Definition: Each term after the first is obtained by multiplying the previous term by a fixed number (common ratio, r)123.

• General term: $a_n=a_1\cdot r^{n-1}$

• Sum of first n terms:
  $S_n=a_1\frac{1-r^n}{1-r}$ (if $r\mathrm{\ne }1$)

• Sum to infinity (|r| < 1):
  $S_{\mathrm{\infty }}=\frac{a_1}{1-r}$

• Example: 2, 6, 18, 54, ... (r = 3)

D. Harmonic Progression (HP)

• Definition: A sequence where the reciprocals form an AP14.

• Example: 1, 1/2, 1/3, 1/4, ...

E. Fibonacci Sequence

• Definition: Each term is the sum of the two preceding terms1.

• Example: 0, 1, 1, 2, 3, 5, 8, 13, ...

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3. Key Concepts and Formulas

• nth term of AP: $a_n=a_1+(n-1)d$

• Sum of n terms of AP: $S_n=\frac{n}{2}[2a_1+(n-1)d]$

• nth term of GP: $a_n=a_1\cdot r^{n-1}$

• Sum of n terms of GP: $S_n=a_1\frac{1-r^n}{1-r}$

• Sum to infinity (GP, |r|<1): $S_{\mathrm{\infty }}=\frac{a_1}{1-r}$

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4. How to Approach Number Series Problems

• Identify the pattern: Look for addition, subtraction, multiplication, division, squares, cubes, or alternating patterns.

• Check for AP/GP/HP/Fibonacci rules.

• For missing/wrong terms: Check differences or ratios between terms.

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5. Examples

Example 1 (AP):
Find the 10th term of 5, 8, 11, ...
$a_{10}=5+(10-1)\times 3=5+27=32$

Example 2 (GP):
Find the sum of first 4 terms of 2, 6, 18, ...
$S_4=2\frac{1-3^4}{1-3}=2\frac{1-81}{-2}=2\times 40=80$

Example 3 (Number Series):
Find the missing term: 1, 4, 9, 16, ?
Pattern: n² ⇒ 1, 2², 3², 4², 5² ⇒ 25

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100 Practice Questions with Answers and Explanations

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Number Series (Missing/Wrong Terms)

1. 2, 4, 8, 16, ?
   Ans: 32 (×2 pattern)

2. 5, 10, 20, 40, ?
   Ans: 80 (×2)

3. 1, 3, 6, 10, 15, ?
   Ans: 21 (+2, +3, +4, ...)

4. 2, 6, 12, 20, ?
   Ans: 30 (+4, +6, +8, ...)

5. 7, 14, 28, 56, ?
   Ans: 112 (×2)

6. 10, 17, 26, 37, ?
   Ans: 50 (+7, +9, +11, ...)

7. 3, 6, 12, 24, ?
   Ans: 48 (×2)

8. 81, 27, 9, 3, ?
   Ans: 1 (÷3)

9. 1, 4, 9, 16, ?
   Ans: 25 (n²)

10. 2, 5, 10, 17, ?
    Ans: 26 (+3, +5, +7, +9)

11. 5, 9, 17, 33, ?
    Ans: 65 (×2 - 1)

12. 100, 81, 64, 49, ?
    Ans: 36 (n², n decreasing)

13. 13, 11, 8, 4, ?
    Ans: -1 (-2, -3, -4, -5)

14. 6, 11, 21, 41, ?
    Ans: 81 (×2 - 1)

15. 1, 2, 6, 24, ?
    Ans: 120 (factorials: 1!, 2!, 3!, 4!, 5!)

16. 2, 4, 7, 11, 16, ?
    Ans: 22 (+2, +3, +4, +5, ...)

17. 8, 16, 32, 64, ?
    Ans: 128 (×2)

18. 1, 8, 27, 64, ?
    Ans: 125 (n³)

19. 21, 18, 15, 12, ?
    Ans: 9 (-3)

20. 5, 6, 11, 17, ?
    Ans: 28 (+1, +5, +6, +11)

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Arithmetic Progression (AP)

21. Find the 12th term of 3, 7, 11, ...
    Ans: 47 (a₁=3, d=4; a₁ₙ=3+(12-1)×4=47)

22. What is the sum of first 20 terms of 2, 5, 8, ...?
    Ans: 620 (n=20, a₁=2, d=3; Sₙ=20/2[2×2+19×3]=10[4+57]=10×61=610)

23. If the 5th term is 20 and common difference is 3, what is the first term?
    Ans: 8 (a₅=a₁+4d=20 ⇒ a₁=20-12=8)

24. Find the sum of first 10 terms of 4, 8, 12, ...
    Ans: 220 (n=10, a₁=4, d=4; Sₙ=5[8+36]=5×44=220)

25. What is the 15th term of 7, 12, 17, ...?
    Ans: 77 (a₁=7, d=5; a₁₅=7+14×5=77)

26. The sum of first n terms of an AP is 100, a₁=2, d=3. Find n.
    Ans: Sₙ=n/2[2×2+(n-1)×3]=100; Solve for n.

27. If a₁=5, d=2, find the 20th term.
    Ans: 43 (a₂₀=5+19×2=43)

28. What is the sum of first 15 terms of 10, 20, 30, ...?
    Ans: 1200 (n=15, a₁=10, d=10; Sₙ=15/2[20+140]=7.5×160=1200)

29. If the 8th term is 50 and d=6, find the first term.
    Ans: 8 (a₈=a₁+7d=50 ⇒ a₁=50-42=8)

30. Find the sum of first 5 terms if a₁=4, d=3.
    Ans: 40 (S₅=5/2[8+12]=2.5×20=50)

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Geometric Progression (GP)

31. Find the 6th term of 2, 6, 18, ...
    Ans: 486 (a₁=2, r=3; a₆=2×3⁵=486)

32. What is the sum of first 4 terms of 3, 6, 12, ...?
    Ans: 45 (S₄=3(1-2⁴)/(1-2)=3(1-16)/-1=3×15=45)

33. If a₁=5, r=2, find the 7th term.
    Ans: 320 (a₇=5×2⁶=320)

34. Find the sum of first 5 terms of 1, 3, 9, ...
    Ans: 121 (S₅=1(1-3⁵)/(1-3)=(1-243)/-2=242/2=121)

35. What is the 8th term of 4, 12, 36, ...?
    Ans: 8748 (a₈=4×3⁷=8748)

36. If the 4th term is 81 and r=3, find the first term.
    Ans: 3 (a₄=a₁×3³=81 ⇒ a₁=81/27=3)

37. Find the sum of first 6 terms if a₁=2, r=2.
    Ans: 126 (S₆=2(1-2⁶)/(1-2)=2(1-64)/-1=2×63=126)

38. What is the sum to infinity of 8, 4, 2, ...?
    Ans: 16 (S∞=8/(1-0.5)=8/0.5=16)

39. If a₁=10, r=0.1, find the sum to infinity.
    Ans: 10/(1-0.1)=10/0.9≈11.11

40. Find the 5th term of 7, 21, 63, ...
    Ans: 567 (a₅=7×3⁴=567)

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Harmonic Progression (HP) & Fibonacci

41. What is the 4th term of the HP: 1, 1/2, 1/3, 1/4, ...?
    Ans: 1/4

42. If the reciprocals of a sequence are 2, 4, 6, 8, ... what is the 3rd term of the HP?
    Ans: 1/6

43. Find the 6th term of the Fibonacci sequence.
    Ans: 8 (0,1,1,2,3,5,8)

44. What is the sum of the first 5 Fibonacci numbers?
    Ans: 0+1+1+2+3=7

45. What is the 7th term of the HP: 1, 1/2, 1/3, ...?
    Ans: 1/7

46. If the sequence is 1, 3, 6, 10, 15, ... what is the 6th term?
    Ans: 21 (triangular numbers)

47. Find the 5th term of the Fibonacci sequence.
    Ans: 5

48. What is the sum of first 4 terms of HP: 1, 1/2, 1/3, 1/4?
    Ans: 1+0.5+0.333+0.25=2.083

49. What is the 8th term of the Fibonacci sequence?
    Ans: 21

50. If the reciprocals are 3, 6, 9, 12, ... what is the 4th term of the HP?
    Ans: 1/12

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Mixed Series & Patterns

51. Find the missing term: 2, 5, 10, 17, ?
    Ans: 26 (+3, +5, +7, +9)

52. Which term is wrong: 2, 4, 8, 16, 33, 64?
    Ans: 33 (should be 32)

53. Find the 7th term of 2, 5, 10, 17, ...
    Ans: 50

54. Find the sum of first 6 terms of 1, 4, 7, ...
    Ans: 1+4+7+10+13+16=51

55. What is the 9th term of 3, 6, 12, ...?
    Ans: 3×2⁸=768

56. Find the sum of first 5 terms of 2, 6, 18, ...
    Ans: 2+6+18+54+162=242

57. Find the missing term: 1, 4, 9, 16, __?, 36
    Ans: 25

58. Which term is wrong: 5, 10, 15, 25, 30?
    Ans: 25 (should be 20)

59. Find the 10th term of 7, 14, 28, ...
    Ans: 7×2⁹=3584

60. Find the sum of first 4 terms of 1, 2, 4, 8, ...
    Ans: 1+2+4+8=15

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Advanced AP/GP Applications

61. If the 1st term is 3 and the 4th term is 15 in an AP, find d.
    Ans: a₄=3+3d=15 ⇒ d=4

62. In a GP, a₁=6, a₄=162. Find r.
    Ans: a₄=6×r³=162 ⇒ r³=27 ⇒ r=3

63. The sum of first n terms of an AP is 210, a₁=5, d=2. Find n.
    Ans: Sₙ=n/2[2×5+(n-1)×2]=210; solve for n.

64. If a₁=2, r=0.5, find the sum to infinity.
    Ans: 2/(1-0.5)=4

65. If the 5th term of an AP is 23 and d=5, find a₁.
    Ans: a₅=a₁+4d=23 ⇒ a₁=3

66. In a GP, a₁=8, r=0.25, find the sum of first 6 terms.
    Ans: S₆=8(1-0.25⁶)/(1-0.25)

67. If the sum of first 10 terms of an AP is 155, a₁=2, find d.
    Ans: S₁₀=10/2[4+9d]=155; solve for d.

68. In a GP, a₁=5, r=2, find the sum of first 5 terms.
    Ans: S₅=5(1-2⁵)/(1-2)=5(1-32)/-1=5×31=155

69. If a₁=7, d=3, find the 15th term.
    Ans: a₁₅=7+14×3=49

70. In a GP, a₁=3, r=4, find the 6th term.
    Ans: 3×4⁵=3072

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Challenging Series and Patterns

71. Find the sum of first 20 terms of 1, 3, 5, ...
    Ans: n=20, a₁=1, d=2; Sₙ=20/2[2+38]=10×40=400

72. Find the 12th term of 5, 10, 20, ...
    Ans: 5×2¹¹=10240

73. Find the sum of first 8 terms of 2, 4, 8, ...
    Ans: S₈=2(1-2⁸)/(1-2)=2(1-256)/-1=2×255=510

74. If a₁=4, d=7, what is the 10th term?
    Ans: 4+9×7=67

75. In a GP, a₁=9, r=1/3, find the sum to infinity.
    Ans: 9/(1-1/3)=13.5

76. Find the missing term: 1, 2, 6, 24, ?
    Ans: 120 (factorials)

77. Which term is wrong: 2, 4, 8, 18, 32?
    Ans: 18 (should be 16)

78. Find the 7th term of 3, 6, 12, ...
    Ans: 3×2⁶=192

79. Find the sum of first 6 terms of 1, 2, 4, 8, ...
    Ans: 1+2+4+8+16+32=63

80. Find the 9th term of 2, 10, 50, ...
    Ans: 2×5⁸=62500

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Mixed Reasoning and Calculation

81. The sum of first n natural numbers is?
    Ans: n(n+1)/2

82. The sum of first n odd numbers is?
    Ans: n²

83. The sum of first n even numbers is?
    Ans: n(n+1)

84. The nth term of the sequence 7, 11, 15, ...?
    Ans: 7+4(n-1)

85. The nth term of 3, 6, 12, ...?
    Ans: 3×2ⁿ⁻¹

86. The sum of first n terms of 5, 10, 15, ...?
    Ans: n/2[10+5(n-1)]

87. The sum of first n terms of 2, 6, 18, ...?
    Ans: 2(1-3ⁿ)/(1-3)

88. Find the 10th term of 1, 3, 9, ...
    Ans: 1×3⁹=19683

89. Find the sum of first 4 terms of 4, 8, 16, ...
    Ans: 4+8+16+32=60

90. The sum of first 5 terms of 1, 2, 3, 4, 5 is?
    Ans: 15

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Higher-Order Reasoning

91. If the sum of first n terms of an AP is 3n²+2n, find the nth term.
    Ans: Sₙ-Sₙ₋₁=3n²+2n-[3(n-1)²+2(n-1)]=6n-1

92. If the sum of first n terms of a GP is 31, a₁=1, r=2, find n.
    Ans: Sₙ=1(1-2ⁿ)/(1-2)=2ⁿ-1=31 ⇒ n=5

93. If the sum of first n terms of an AP is n(n+1), what is the common difference?
    Ans: Sₙ=n(n+1), so a₁=2, d=1

94. If the sum of first n terms of a GP is 120, a₁=3, r=2, find n.
    Ans: Sₙ=3(1-2ⁿ)/(1-2)=3×(2ⁿ-1)=120 ⇒ 2ⁿ=41 ⇒ n≈5.36 (not integer)

95. If the sum of first n terms of an AP is 5n, what is the nth term?
    Ans: Sₙ-Sₙ₋₁=5

96. If a₁=2, d=3, what is the 25th term?
    Ans: 2+24×3=74

97. If a₁=1, r=0.5, find the sum to infinity.
    Ans: 1/(1-0.5)=2

98. If the 3rd term of a GP is 24, a₁=3, find r.
    Ans: 3×r²=24 ⇒ r²=8 ⇒ r=2√2

99. If the sum of first n terms of an AP is 2n², what is the nth term?
    Ans: Sₙ-Sₙ₋₁=4n-2

100. If the sum of first n terms of a GP is 21, a₁=1, r=2, find n.
     Ans: 2ⁿ-1=21 ⇒ n=5

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Use these notes and questions to master sequences and series, including number series, AP, GP, HP, and pattern recognition, for all competitive exams!
For more details and solved examples, refer to standard math resources and the referenced sites above.