Chapter 5: Arithmetic Progressions

Class 10 Mathematics · MCQs with answers

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What is the common difference of the AP 3, 1, −1, −3, …?
1. (A) 3
2. (B) 2
3. (C) −1
4. (D) −2
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Correct answer: (D) −2

Explanation: Common difference d = second term − first term = 1 − 3 = −2. Each term is 2 less than the one before.

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Which of these is an arithmetic progression (AP)?
1. (A) 1, 2, 4, 8, …
2. (B) 2, 4, 6, 8, …
3. (C) 1, 4, 9, 16, …
4. (D) 1, 1/2, 1/3, 1/4, …
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Correct answer: (B) 2, 4, 6, 8, …

Explanation: In an AP, the difference between consecutive terms stays the same. In 2, 4, 6, 8, … it is always 2; in the other sequences it changes.

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What is the nth term of an AP with first term a and common difference d?
1. (A) a + nd
2. (B) a − (n − 1)d
3. (C) n + (a − 1)d
4. (D) a + (n − 1)d
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Correct answer: (D) a + (n − 1)d

Explanation: The nth term is aₙ = a + (n − 1)d. The common difference is added to the first term (n − 1) times.

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What is the 10th term of the AP 2, 7, 12, …?
1. (A) 37
2. (B) 42
3. (C) 45
4. (D) 47
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Correct answer: (D) 47

Explanation: a = 2 and d = 5. a₁₀ = 2 + 9 × 5 = 47.

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Which term of the AP 21, 18, 15, … is −81?
1. (A) 35th
2. (B) 34th
3. (C) 36th
4. (D) 33rd
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Correct answer: (A) 35th

Explanation: a = 21 and d = −3. 21 + (n − 1)(−3) = −81 gives (n − 1) × 3 = 102, so n − 1 = 34 and n = 35.

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Is 301 a term of the AP 5, 11, 17, 23, …?
1. (A) Yes, the 49th term
2. (B) Yes, the 50th term
3. (C) Yes, the 51st term
4. (D) No, 301 is not a term of this AP
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Correct answer: (D) No, 301 is not a term of this AP

Explanation: 5 + (n − 1) × 6 = 301 gives n − 1 = 296/6, which is not a whole number. n must be a whole number, so 301 is not a term of this AP.

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How many two-digit numbers are divisible by 3?
1. (A) 33
2. (B) 31
3. (C) 30
4. (D) 29
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Correct answer: (C) 30

Explanation: The numbers are 12, 15, …, 99: a = 12, d = 3, aₙ = 99. 12 + (n − 1)3 = 99 gives n − 1 = 29, so n = 30.

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What is the 11th term from the last of the AP 10, 7, 4, …, −62?
1. (A) −38
2. (B) −35
3. (C) −32
4. (D) −29
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Correct answer: (C) −32

Explanation: Counting from the end gives the AP −62, −59, …, with a = −62 and d = 3. The 11th term = −62 + 10 × 3 = −32.

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What is the sum of the first 100 natural numbers?
1. (A) 10100
2. (B) 5050
3. (C) 5000
4. (D) 4950
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Correct answer: (B) 5050

Explanation: The sum of the first n natural numbers is n(n + 1)/2: 100 × 101 / 2 = 5050.

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What is the formula for Sₙ, the sum of the first n terms of an AP?
1. (A) Sₙ = n [2a + (n − 1)d]
2. (B) Sₙ = n/2 [2a + (n − 1)d]
3. (C) Sₙ = n/2 [a + (n − 1)d]
4. (D) Sₙ = 2n [a + (n − 1)d]
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Correct answer: (B) Sₙ = n/2 [2a + (n − 1)d]

Explanation: Sₙ = n/2 [2a + (n − 1)d]. If the last term l is known, Sₙ = n/2 (a + l) also works.

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What is the sum of the first 22 terms of the AP 8, 3, −2, …?
1. (A) 979
2. (B) −869
3. (C) −979
4. (D) −1089
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Correct answer: (C) −979

Explanation: a = 8 and d = −5. S₂₂ = 22/2 [16 + 21 × (−5)] = 11 × (16 − 105) = 11 × (−89) = −979.

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What is the sum of the first 15 multiples of 8?
1. (A) 120
2. (B) 900
3. (C) 960
4. (D) 1200
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Correct answer: (C) 960

Explanation: 8 + 16 + … + 120 = 8 × (1 + 2 + … + 15) = 8 × 120 = 960.

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What is the sum of all odd numbers between 0 and 50?
1. (A) 1250
2. (B) 650
3. (C) 625
4. (D) 600
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Correct answer: (C) 625

Explanation: 1, 3, …, 49 has a = 1, d = 2 and n = 25 terms. S = 25/2 × (1 + 49) = 25 × 25 = 625.

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If 2x, x + 10 and 3x + 2 are in AP, what is x?
1. (A) 2
2. (B) 3
3. (C) 4
4. (D) 6
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Correct answer: (D) 6

Explanation: In an AP, the middle term is the average of its neighbours: 2(x + 10) = 2x + 3x + 2, so 2x + 20 = 5x + 2 and x = 6. (The terms are 12, 16, 20.)

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If the nth term of an AP is aₙ = 3 + 4n, what is the common difference?
1. (A) 11
2. (B) 7
3. (C) 4
4. (D) 3
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Correct answer: (C) 4

Explanation: a₁ = 7, a₂ = 11, a₃ = 15, …. Each term is 4 more than the previous one, so d = 4 (the coefficient of n).

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The sum of the first n terms of an AP is Sₙ = 4n − n². What is its 10th term?
1. (A) −15
2. (B) −13
3. (C) −11
4. (D) 15
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Correct answer: (A) −15

Explanation: aₙ = Sₙ − Sₙ₋₁. a₁₀ = S₁₀ − S₉ = (40 − 100) − (36 − 81) = −60 + 45 = −15.

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How many terms are there in the AP 7, 13, 19, …, 205?
1. (A) 34
2. (B) 33
3. (C) 30
4. (D) 28
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Correct answer: (A) 34

Explanation: a = 7 and d = 6. 7 + (n − 1) × 6 = 205 gives n − 1 = 33, so n = 34.

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The 11th term of an AP is 38 and the 16th term is 73. What is its 31st term?
1. (A) 178
2. (B) 185
3. (C) 188
4. (D) 192
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Correct answer: (A) 178

Explanation: a + 10d = 38 and a + 15d = 73 give 5d = 35, so d = 7 and a = −32. a₃₁ = −32 + 30 × 7 = 178.

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An AP has a = 5, d = 3 and aₙ = 50. What is n?
1. (A) 18
2. (B) 17
3. (C) 16
4. (D) 15
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Correct answer: (C) 16

Explanation: 5 + (n − 1) × 3 = 50 gives n − 1 = 15, so n = 16.

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Ramkali saved ₹5 in the first week and then increased her weekly savings by ₹1.75 each week. In which week will her weekly savings be ₹20.75?
1. (A) In the 9th week
2. (B) In the 11th week
3. (C) In the 10th week
4. (D) In the 12th week
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Correct answer: (C) In the 10th week

Explanation: a = 5 and d = 1.75. 5 + (n − 1) × 1.75 = 20.75 gives (n − 1) × 1.75 = 15.75, so n − 1 = 9 and n = 10.

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₹700 is to be given as seven cash prizes. Each prize is ₹20 less than the one before it. What is the largest prize?
1. (A) ₹160
2. (B) ₹140
3. (C) ₹120
4. (D) ₹100
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Correct answer: (A) ₹160

Explanation: S₇ = 7/2 [2a + 6 × (−20)] = 700 gives 2a − 120 = 200, so a = 160. The prizes are ₹160, ₹140, …, ₹40.

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If the first term of an AP is a and the last term is l, what is the sum of its n terms?
1. (A) n (a + l)
2. (B) (a + l)/n
3. (C) n/2 (a − l)
4. (D) n/2 (a + l)
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Correct answer: (D) n/2 (a + l)

Explanation: The average of the first and last terms is (a + l)/2. Multiplying it by the number of terms n gives the sum n/2 (a + l).

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Which term of the AP 3, 15, 27, 39, … is 132 more than its 54th term?
1. (A) 64th
2. (B) 65th
3. (C) 66th
4. (D) 55th
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Correct answer: (B) 65th

Explanation: d = 12. 132 ÷ 12 = 11, so the term is 11 places after the 54th: 54 + 11 = 65th. (a₅₄ = 639 and a₆₅ = 771.)

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What is the common difference of the AP 5, 5, 5, 5, …?
1. (A) −10
2. (B) −5
3. (C) −1
4. (D) 0
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Correct answer: (D) 0

Explanation: All the terms are equal, so the difference between consecutive terms is 5 − 5 = 0. This is an AP with common difference 0.

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Assertion (A): The sequence 1, 4, 9, 16, … is not an AP. Reason (R): The differences between its consecutive terms (3, 5, 7, …) are not equal.
1. (A) Both A and R are true, and R is the correct explanation of A.
2. (B) Both A and R are true, but R is not the correct explanation of A.
3. (C) A is true, but R is false.
4. (D) A is false, but R is true.
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Correct answer: (A) Both A and R are true, and R is the correct explanation of A.

Explanation: For an AP, consecutive terms must differ by the same amount. Here the differences keep changing, so it is not an AP, and R correctly explains A.

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