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About Arithmetic Progressions — Class 10 CBSE

Find nth term and sum of n terms of an arithmetic progression with real-life applications. This topic is part of the CBSE Class 10 mathematics syllabus (chapter: Chapter 5). On this page you can practice 62 questions across three difficulty levels — 21 easy, 20 medium, and 21 hard — each with a visual step-by-step solution, plus a timed 36-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

What you'll learn in Arithmetic Progressions

  • Introduction to Arithmetic Progressions (AP)
  • The $n^{th}$ Term of an AP
  • Applications of the $n^{th}$ Term Formula
  • Sum of $n$ Terms of an AP
  • Advanced Applications and Problem Solving

Interactive lesson · about 15 minutes · checkpoint question after every unit

Arithmetic Progressions — solved examples for Class 10 CBSE

Example 1easy

Which of the following sequences is an Arithmetic Progression (AP)?
  1. A)2, 4, 8, 16, ...
  2. B)1, 4, 9, 16, ...
  3. C)3, 6, 9, 12, ...
  4. D)1/2, 1/3, 1/4, 1/5, ...

Step-by-step solution

  1. For an AP, the common difference 'd' must be constant for all consecutive terms.
  2. Option A: 4-2=2, 8-4=4. Not an AP.
  3. Option B: 4-1=3, 9-4=5. Not an AP.
  4. Option C: 6-3=3, 9-6=3, 12-9=3. The common difference is constant (d=3), so it is an AP.
  5. Option D: 1/3 - 1/2 = -1/6, 1/4 - 1/3 = -1/12. Not an AP.

Answer: 3, 6, 9, 12, ...

Example 2medium

Which of the following sequences is an Arithmetic Progression (AP)?
  1. A)1, 3, 6, 10, ...
  2. B)2, 4, 8, 16, ...
  3. C)3, 6, 9, 12, ...
  4. D)1/2, 1/3, 1/4, 1/5, ...

Step-by-step solution

  1. For a sequence to be an AP, the common difference 'd' (a_n - a_{n-1}) must be constant for all terms.
  2. Option A: 3-1=2, 6-3=3. Not an AP.
  3. Option B: 4-2=2, 8-4=4. Not an AP (this is a Geometric Progression).
  4. Option C: 6-3=3, 9-6=3, 12-9=3. The common difference is constant (d=3). So, this is an AP.
  5. Option D: 1/3 - 1/2 = -1/6, 1/4 - 1/3 = -1/12. Not an AP.

Answer: 3, 6, 9, 12, ...

Example 3hard

If the pth term of an Arithmetic Progression is 1/q and the qth term is 1/p, where p ≠ q, what is the (pq)th term of this AP?
  1. A)0
  2. B)1/(pq)
  3. C)1
  4. D)p+q

Step-by-step solution

  1. Given ap = a + (p-1)d = 1/q and aq = a + (q-1)d = 1/p.
  2. Subtracting the two equations, we get (p-q)d = 1/q - 1/p = (p-q)/(pq). Since p ≠ q, d = 1/(pq).
  3. Substitute d = 1/(pq) into a + (p-1)d = 1/q: a + (p-1)/(pq) = 1/q. Solving for a, we get a = 1/q - (p-1)/(pq) = (p - (p-1))/(pq) = 1/(pq).
  4. The (pq)th term is a + (pq-1)d = 1/(pq) + (pq-1)/(pq) = (1 + pq - 1)/(pq) = pq/(pq) = 1.

Answer: 1

Practice questions on Arithmetic Progressions

  1. Q1.easy

    For an AP, if the first term is 'a' and the common difference is 'd', which of the following statements is INCORRECT?
    1. A)The n-th term is given by a_n = a + (n-1)d
    2. B)The common difference 'd' can be zero.
    3. C)The terms of an AP always increase if 'd' is positive.
    4. D)If a sequence has a constant difference between consecutive terms, it is an AP.
    Show answer

    Answer: The terms of an AP always increase if 'd' is positive.

    Hint: Consider what happens to the terms if the common difference is negative. Also, what if 'd' is zero?

  2. Q2.easy

    A student wrote the first few terms of an AP as 5, 8, 11, 14, ... and concluded that the 10th term is 32. Identify the mistake in their reasoning.
    1. A)The first term 'a' is incorrectly identified.
    2. B)The common difference 'd' is incorrectly calculated.
    3. C)The formula for the n-th term was applied incorrectly.
    4. D)There is no mistake; the 10th term is indeed 32.
    Show answer

    Answer: The formula for the n-th term was applied incorrectly.

    Hint: Carefully apply the formula for the n-th term, a_n = a + (n-1)d. Pay attention to the 'n-1' part.

  3. Q3.easy

    If the common difference of an AP is 5, what is the value of a_18 - a_13?
    1. A)5
    2. B)25
    3. C)15
    4. D)30
    Show answer

    Answer: 25

    Hint: Recall the formula for the n-th term and how it relates to the first term and common difference. You can express both a_18 and a_13 using this formula.

  4. Q4.medium

    The first term of an AP is 5 and the common difference is 3. What is its 15th term?
    1. A)45
    2. B)47
    3. C)50
    4. D)52
    Show answer

    Answer: 47

    Hint: Recall the formula for the nth term of an AP: a_n = a + (n-1)d. Substitute the given values carefully.

  5. Q5.medium

    In an AP, if the 3rd term is 12 and the 7th term is 24, what is the common difference?
    1. A)2
    2. B)3
    3. C)4
    4. D)6
    Show answer

    Answer: 3

    Hint: Set up two equations using the nth term formula for the 3rd and 7th terms. Then solve the system of equations to find 'd'.

  6. Q6.medium

    How many terms are there in the AP: 7, 13, 19, ..., 205?
    1. A)33
    2. B)34
    3. C)35
    4. D)36
    Show answer

    Answer: 34

    Hint: Identify the first term, common difference, and the last term. Use the formula for the nth term to find 'n'.

  7. Q7.hard

    A man decides to save money. He saves ₹100 in the first month. In each subsequent month, he saves ₹50 more than he did in the previous month. If his total savings at the end of 'n' months is ₹19000, and his savings in the 10th month is 'X', find the value of (X/100).
    1. A)19
    2. B)5.5
    3. C)10
    4. D)550
    Show answer

    Answer: 5.5

    Hint: First, use the sum formula for an AP to find the number of months 'n'. Then, calculate the savings in the 10th month (X) using the nth term formula.

  8. Q8.hard

    If the sum of the first 'n' terms of an Arithmetic Progression is given by the formula Sn = 3n^2 + 5n, find the common difference of the AP.
    1. A)3
    2. B)4
    3. C)6
    4. D)8
    Show answer

    Answer: 6

    Hint: The nth term of an AP can be found using the relation an = Sn - S(n-1). Calculate the first two terms and then find the common difference.

  9. Q9.hard

    If the ratio of the 11th term to the 18th term of an AP is 2:3, what is the ratio of its 5th term to its 21st term?
    1. A)1/3
    2. B)2/3
    3. C)1/4
    4. D)2/5
    Show answer

    Answer: 1/3

    Hint: Express the 11th and 18th terms using 'a' and 'd'. Use their ratio to find a relationship between 'a' and 'd'. Then, apply this relationship to find the ratio of the 5th and 21st terms.

These are 9 of the 62 questions available for Arithmetic Progressions. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.