1. The nth term of an AP is given by aₙ = a + (n - 1)d. Here, 'd' represents:
A. First term
B. Common ratio
C. Common difference
D. Number of terms
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Answer: C. Common difference
Explanation: In an Arithmetic Progression, 'a' is the first term, 'n' is the number of terms, and 'd' is the common difference.
2. What is the common difference of the AP: 10, 7, 4, 1, ...?
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Answer: B. -3
Explanation: Common difference d = a₂ - a₁ = 7 - 10 = -3.
3. The 10th term of the AP: 2, 7, 12, ... is:
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Answer: B. 47
Explanation: a = 2, d = 5, n = 10. a₁₀ = 2 + (10 - 1) × 5 = 2 + 45 = 47.
4. Which term of the AP: 21, 18, 15, ... is 0?
A. 6th term
B. 7th term
C. 8th term
D. 9th term
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Answer: C. 8th term
Explanation: a = 21, d = -3, aₙ = 0. 0 = 21 + (n - 1)(-3) ⇒ -21 = -3(n - 1) ⇒ n - 1 = 7 ⇒ n = 8.
5. The sum of first n terms of an AP is given by the formula:
A. Sₙ = n/2 [a + l]
B. Sₙ = n/2 [2a + (n - 1)d]
C. Both A and B
D. Sₙ = n [a + (n - 1)d]
Show Answer
Answer: C. Both A and B
Explanation: Both Sₙ = n/2 [2a + (n - 1)d] and Sₙ = n/2 [a + l] (where l is last term) are valid formulas for the sum of an AP.
6. The 11th term of the AP: -3, -1/2, 2, ... is:
A. 28
B. 22
C. -38
D. -46.5
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Answer: B. 22
Explanation: a = -3, d = -1/2 - (-3) = 2.5 = 5/2. a₁₁ = -3 + (10)(5/2) = -3 + 25 = 22.
7. Sum of the first 10 natural numbers is:
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Answer: C. 55
Explanation: Sₙ = n(n + 1)/2. For n = 10, S₁₀ = 10 × 11 / 2 = 55.
8. If the first term of an AP is 5 and the common difference is 3, then the 5th term is:
Show Answer
Answer: A. 17
Explanation: a₅ = a + 4d = 5 + 4(3) = 5 + 12 = 17.
9. What is the sum of the first 100 positive integers?
A. 5050
B. 5005
C. 5000
D. 5500
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Answer: A. 5050
Explanation: S₁₀₀ = 100 × 101 / 2 = 5050.
10. If k, 2k - 1, 2k + 1 are three consecutive terms of an AP, then the value of k is:
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Answer: B. 3
Explanation: (2k - 1) - k = (2k + 1) - (2k - 1) ⇒ k - 1 = 2 ⇒ k = 3.
11. The common difference of an AP can be:
A. Positive only
B. Negative only
C. Zero only
D. Positive, negative, or zero
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Answer: D. Positive, negative, or zero
Explanation: The common difference 'd' can be any real number: positive, negative, or zero.
12. The nth term of an AP is 3n + 5. The common difference is:
Show Answer
Answer: A. 3
Explanation: a₁ = 3(1) + 5 = 8, a₂ = 3(2) + 5 = 11. Common difference d = 11 - 8 = 3. (Coefficient of n is always d).
13. The 4th term from the end of the AP: -11, -8, -5, ..., 49 is:
Show Answer
Answer: B. 40
Explanation: Reverse the AP: 49, 46, 43, ... Here first term L = 49, common difference d' = -3. 4th term = 49 + 3(-3) = 49 - 9 = 40.
14. If a = 7, d = 3, n = 8, then aₙ is:
Show Answer
Answer: A. 28
Explanation: a₈ = 7 + (8 - 1) × 3 = 7 + 21 = 28.
15. How many two-digit numbers are divisible by 3?
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Answer: B. 30
Explanation: Two-digit numbers divisible by 3 are 12, 15, ..., 99. Here a = 12, d = 3, last term l = 99. 99 = 12 + (n - 1)3 ⇒ 87 = 3(n - 1) ⇒ n = 30.
16. The sum of first n odd natural numbers is:
A. 2n - 1
B. 2n + 1
C. n²
D. n² + 1
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Answer: C. n²
Explanation: AP: 1, 3, 5, ... Sum Sₙ = n/2 [2(1) + (n - 1)2] = n/2 [2n] = n².
17. The sum of first n even natural numbers is:
A. n²
B. n(n + 1)
C. n(n - 1)
D. 2n²
Show Answer
Answer: B. n(n + 1)
Explanation: AP: 2, 4, 6, ... Sum Sₙ = n/2 [2(2) + (n - 1)2] = n/2 [2n + 2] = n(n + 1).
18. If the sum of first n terms of an AP is 3n² + 5n, its 2nd term is:
Show Answer
Answer: B. 14
Explanation: S₁ = 3(1) + 5 = 8 = a₁. S₂ = 3(4) + 5(2) = 22. Second term a₂ = S₂ - S₁ = 22 - 8 = 14.
19. If a = -18, n = 10, a = 0, then common difference d is:
Show Answer
Answer: A. 2
Explanation: 0 = -18 + (10 - 1)d ⇒ 9d = 18 ⇒ d = 2.
20. Which of the following sequence is an AP?
A. 1, 3, 9, 27, ...
B. 2, 4, 8, 16, ...
C. √2, √8, √18, √32, ...
D. 1, 2, 4, 7, 11, ...
Show Answer
Answer: C. √2, √8, √18, √32, ...
Explanation: √2, 2√2, 3√2, 4√2, ... has a constant common difference of √2.
21. The common difference of the AP: √3, √12, √27, √48, ... is:
Show Answer
Answer: A. √3
Explanation: Terms are √3, 2√3, 3√3, 4√3. Common difference d = 2√3 - √3 = √3.
22. If 18, a, b, -3 are in AP, then a + b =
Show Answer
Answer: B. 15
Explanation: First term = 18, 4th term = -3 ⇒ 18 + 3d = -3 ⇒ 3d = -21 ⇒ d = -7. So a = 18 - 7 = 11 and b = 11 - 7 = 4. Thus a + b = 11 + 4 = 15.
23. In an AP, if d = -4, n = 7, aₙ = 4, then a is:
Show Answer
Answer: D. 28
Explanation: 4 = a + (7 - 1)(-4) ⇒ 4 = a - 24 ⇒ a = 28.
24. The famous mathematician associated with finding the sum of first 100 natural numbers as a child is:
A. Pythagoras
B. Gauss
C. Euclid
D. Newton
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Answer: B. Gauss
Explanation: Carl Friedrich Gauss discovered the method for summing an arithmetic series at a very young age.
25. If the nth term of an AP is aₙ = 9 - 5n, then its common difference is:
Show Answer
Answer: B. -5
Explanation: Coefficient of n in a linear expression for nth term is the common difference, which is -5.
26. Total number of terms in the AP: 7, 13, 19, ..., 205 is:
Show Answer
Answer: B. 34
Explanation: 205 = 7 + (n - 1)6 ⇒ 198 = 6(n - 1) ⇒ n - 1 = 33 ⇒ n = 34.
27. If the 3rd and 9th terms of an AP are 4 and -8 respectively, which term is 0?
A. 4th term
B. 5th term
C. 6th term
D. 7th term
Show Answer
Answer: B. 5th term
Explanation: a₃ = a + 2d = 4, a₉ = a + 8d = -8. Subtracting gives 6d = -12 ⇒ d = -2, a = 8. For aₙ = 0: 8 + (n - 1)(-2) = 0 ⇒ n = 5.
28. The sum of first 15 multiples of 8 is:
A. 960
B. 1000
C. 880
D. 1040
Show Answer
Answer: A. 960
Explanation: AP: 8, 16, 24, ... up to 15 terms. S₁₅ = 15/2 [2(8) + (14)(8)] = 15/2 [16 + 112] = 15/2 [128] = 960.
29. Find the sum of all 3-digit natural numbers which are multiples of 7 starting from 105 to 994.
A. 70336
B. 70400
C. 69336
D. 70000
Show Answer
Answer: A. 70336
Explanation: First 3-digit multiple of 7 is 105, last is 994. Number of terms n = 128. S₁₂₈ = 128/2 [105 + 994] = 64 × 1099 = 70336.
30. If a, b, c are in AP, then the relation between them is:
A. b = a + c
B. 2b = a + c
C. b² = ac
D. a + b = c
Show Answer
Answer: B. 2b = a + c
Explanation: Common difference d = b - a = c - b ⇒ 2b = a + c (b is the arithmetic mean of a and c).
