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CBSE Class 10 Mathematics Chapter 5 Arithmetic Progressions Worksheet with Answers

Q: How many marks does Arithmetic Progressions carry in CBSE Class 10 board exam?

Chapter 5 Arithmetic Progressions typically carries 6 marks in the CBSE Class 10 Mathematics board exam. This usually comprises one 2-mark short-answer question, one 3-mark question, and sometimes part of a 4-mark case-study question. The chapter is considered moderately scoring if formulas are applied correctly.

Q: What are the most important formulas for nth term and sum of n terms?

The two critical formulas are: (1) nth term aₙ = a + (n-1)d where a is first term, d is common difference, n is term number; (2) Sum of n terms Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2[a + l] where l is last term. Memorize both forms of the sum formula as different problems suit different forms.

Q: How do I identify whether a given sequence is an Arithmetic Progression?

Calculate the difference between consecutive terms. If the difference remains constant throughout, the sequence is an AP. For example, in 5,9,13,17, the differences are 4,4,4 (constant), so it is an AP. If differences vary like in 2,4,7,11, it is not an AP. Alternatively, check if aₙ₊₁ - aₙ is the same for all n.

Q: What is the common mistake students make in word problems on Arithmetic Progressions?

The most frequent error is incorrect identification of the first term 'a' and common difference 'd' from the problem statement. Students often confuse which quantity represents a and which represents d. Always list out at least the first three terms explicitly from the word problem before applying formulas. Also, carefully note whether the problem asks for the term number or the term value itself.

Q: How should I approach case-study questions on Arithmetic Progressions?

Read the entire scenario twice and underline all numerical data. Identify which quantities form the AP — often real-world contexts like seat arrangements, salary increments, or construction schedules. Extract the first term and common difference from the context, then treat sub-questions as standard AP problems using appropriate formulas. The first sub-question usually tests direct formula application while later ones require equation formation.

Q: Is it necessary to learn the derivation of sum of n terms formula?

While CBSE board exams rarely ask for derivation directly, understanding how Sₙ = n/2[2a + (n-1)d] is derived helps you remember the formula correctly and apply it confidently. The derivation using the method of writing the sum forwards and backwards is elegant and worth knowing. However, prioritize formula application practice over derivation memorization for board exam scoring.

Q: How much time should I spend on this worksheet for effective practice?

Allocate 90 minutes for the first attempt under timed conditions to simulate exam pressure. If you are practising for the first time, you may extend to 120 minutes. After completing, spend another 30-40 minutes reviewing the answer key and understanding mistakes. Re-attempt incorrect questions after 2-3 days. Completing 3-4 such full-length worksheets ensures thorough chapter mastery.

Q: What are the easiest marks to score in Arithmetic Progressions chapter?

MCQs and fill-in-the-blanks based on direct formula substitution are the easiest marks. Questions asking for the nth term when first term and common difference are directly given, or sum calculations with all values provided, require only careful arithmetic. Focus on accuracy in these sections first before attempting complex word problems. These objective questions typically constitute 3-4 marks and should be attempted with 100% accuracy target.

Q: Can Arithmetic Progressions formulas be used for geometric sequences?

No, AP formulas apply only when the difference between consecutive terms is constant. Geometric Progressions (GP) have a constant ratio between consecutive terms and require different formulas. Mixing these concepts is a common error. If a problem mentions percentage increase or multiplicative growth, it likely involves GP not AP. Always verify whether common difference or common ratio is constant before selecting formulas.

Q: How does CBSETUTOR.ai help if I get stuck on a specific AP problem at night?

CBSETUTOR.ai operates 24×7, so you can photograph your problem using your smartphone and upload it anytime — even at 11 PM while revising. The AI tutor provides step-by-step solutions within seconds, explaining which formula to use and why. This eliminates waiting until next day for teacher or tutor help. The platform costs just ₹999/month for unlimited access across all subjects and classes 6-12, with a 3-day free trial to test it first.

Arithmetic Progressions is a scoring chapter in CBSE Class 10 Mathematics, typically carrying 6-8 marks in the board exam. This printable worksheet helps students master the nth term formula, sum of n terms, and real-world applications through structured practice. Each section progressively builds difficulty, ensuring thorough concept clarity and exam readiness.

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Key takeaways

✓Practice 30+ questions covering nth term formulas, sum of n terms, and word problems from Arithmetic Progressions
✓Worksheet includes 6 MCQs, 5 fill-in-the-blanks, 5 true/false, 5 short answers, 3 long answers, and 1 case-study question
✓Difficulty level: Moderate to Challenging — suitable for board exam preparation and concept mastery
✓Suggested time: 90 minutes under timed conditions to simulate actual exam environment
✓Complete answer key provided with brief explanations for every question to enable self-study
✓Aligned strictly with NCERT Class 10 Mathematics syllabus and latest CBSE marking scheme
✓Covers all three major topics: nth term calculation, sum of n terms, and application-based word problems

Quick Chapter Recap: Arithmetic Progressions Essentials

An Arithmetic Progression (AP) is a sequence where each term after the first is obtained by adding a fixed number called the common difference (d) to the preceding term. For an AP with first term 'a' and common difference 'd', the nth term is given by aₙ = a + (n - 1)d. The sum of first n terms is Sₙ = n/2[2a + (n - 1)d] or Sₙ = n/2[a + l] where l is the last term. These formulas form the backbone of solving all problems in this chapter. Understanding when to use which formula and how to extract given information from word problems are crucial skills tested in CBSE board exams.

General form of AP: a, a+d, a+2d, a+3d,... where 'a' is first term and 'd' is common difference
nth term formula: aₙ = a + (n - 1)d — use when finding any specific term position
Sum of n terms: Sₙ = n/2[2a + (n - 1)d] or Sₙ = n/2[first term + last term]
If a, b, c are in AP, then b - a = c - b, i.e., 2b = a + c
Common difference d = aₙ - aₙ₋₁ for any consecutive terms

Worksheet Instructions and Time Management

This worksheet is designed for 90 minutes of focused practice under exam-like conditions. Allocate 15 minutes for Section A (MCQs), 10 minutes for Section B (fill-in-the-blanks), 8 minutes for Section C (true/false), 25 minutes for Section D (short answers), 25 minutes for Section E (long answers), and 7 minutes for the case-study question. Keep the remaining time for review. Use a separate answer sheet and show all working clearly for sections D and E to gain full marks. Attempt all questions as negative marking does not apply. Cross-check your answers with the answer key only after completing the entire worksheet to identify weak areas needing revision.

Difficulty Level: Moderate to Challenging — aligned with board exam pattern
Total Time: 90 minutes (suggested), extendable to 120 minutes for first attempt
Materials needed: Calculator (non-programmable), ruler, answer sheet, rough work pages
Marking scheme follows CBSE guidelines: MCQs 1 mark each, Short answers 2-3 marks, Long answers 4-5 marks
Write step-by-step solutions for full credit; partial marks awarded for correct method even if final answer is wrong

Section A: Multiple Choice Questions (1 mark each)

This section tests your ability to quickly identify correct concepts and apply formulas accurately. Each MCQ has four options with only one correct answer. Read each question carefully, especially numerical values and what is being asked. Eliminate obviously wrong options first. Show brief working in the margin if needed, but mark only the final answer clearly. These questions typically cover direct formula application, identification of AP sequences, and basic conceptual understanding of nth term and sum of n terms.

Q1. The 10th term of the AP: 5, 8, 11, 14,... is: (a) 32 (b) 35 (c) 38 (d) 41
Q2. If the common difference of an AP is 5 and the 7th term is 34, then the first term is: (a) 4 (b) 5 (c) 3 (d) 6
Q3. The sum of first 20 terms of the AP: 1, 4, 7, 10,... is: (a) 580 (b) 590 (c) 600 (d) 610
Q4. Which term of the AP: 21, 18, 15,... is –30? (a) 16th (b) 17th (c) 18th (d) 19th
Q5. If 7 times the 7th term of an AP is equal to 11 times its 11th term, then the 18th term is: (a) 0 (b) 7 (c) 11 (d) 18
Q6. The first and last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there? (a) 36 (b) 37 (c) 38 (d) 39

Section B: Fill in the Blanks (1 mark each)

Complete each statement with the correct numerical value or algebraic expression. These questions test recall of formulas, properties of AP, and ability to perform quick mental calculations. Write only the answer in the blank space provided; no working is required for this section. Pay attention to units and signs (positive/negative). Some blanks may require you to state a formula or a term name rather than a number. Ensure your handwriting is clear as ambiguous answers may not receive credit during evaluation.

Q7. The common difference of the AP: 1/3, 1/2, 2/3, 5/6,... is __________.
Q8. If the first term of an AP is –5 and common difference is 2, then the sum of first 10 terms is __________.
Q9. The 20th term from the end of the AP: 3, 8, 13,..., 253 is __________.
Q10. If the sum of first n terms of an AP is 3n² + 5n, then the common difference is __________.
Q11. Three numbers in AP have sum 21 and product 231. The middle number is __________.
Q12. In an AP, if a = 7, d = 3, and aₙ = 52, then n = __________.

Section C: True or False (1 mark each)

Evaluate each statement and write 'True' or 'False'. These questions assess your conceptual clarity and ability to identify common misconceptions in Arithmetic Progressions. Read each statement word-by-word; sometimes a single term can make a statement false. If a statement is false, you should be able to provide a counter-example mentally. No justification is required in your answer sheet, but understanding why each statement is true or false deepens concept mastery. Pay special attention to edge cases and conditions mentioned in the statements.

Q13. Every sequence is an Arithmetic Progression.
Q14. If the common difference of an AP is zero, all terms are equal.
Q15. The sum of first n natural numbers is n(n+1)/2.
Q16. An AP can have negative common difference.
Q17. If three numbers are in AP, their squares are also in AP.

Section D: Short Answer Questions (2-3 marks each)

These questions require you to show your working step-by-step and arrive at the correct answer with proper reasoning. Each question typically involves applying one or two formulas and performing algebraic manipulation or arithmetic calculation. Start by identifying what is given (a, d, n, aₙ, or Sₙ) and what needs to be found. Write the appropriate formula, substitute values, and solve systematically. Marks are awarded for correct method even if calculation errors occur, so never skip steps. Underline or box your final answer for clarity. Aim to complete each question in 4-5 minutes.

Q18. Find the 31st term of an AP whose 11th term is 38 and 16th term is 73. (2 marks)
Q19. How many two-digit numbers are divisible by 3? (2 marks)
Q20. The sum of 4th and 8th terms of an AP is 24 and the sum of 6th and 10th terms is 44. Find the first three terms of the AP. (3 marks)
Q21. Which term of the AP: 3, 15, 27, 39,... will be 132 more than its 54th term? (3 marks)
Q22. Find the sum of all three-digit natural numbers which are divisible by 7. (3 marks)

Section E: Long Answer and HOTS Questions (4-5 marks each)

Long-answer questions test your ability to handle multi-step problems, apply multiple concepts together, and reason logically through complex scenarios. These questions often combine nth term and sum formulas or require you to form and solve simultaneous equations. HOTS (Higher Order Thinking Skills) questions demand analytical thinking and sometimes unconventional approaches. Read the question at least twice to fully grasp what is required. Plan your solution pathway before you start writing. Clearly label equations and steps. Show all algebraic work and arithmetic calculations. These questions typically take 7-9 minutes each. Quality of presentation matters for full marks.

Q23. The sum of first n terms of an AP is given by Sₙ = 2n² + 5n. Find the nth term and show that the sequence is an AP. Also find the 20th term. (4 marks)
Q24. A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹200 for the first day, ₹250 for the second day, ₹300 for the third day, etc., the penalty for each succeeding day being ₹50 more than for the preceding day. How much money the contractor has to pay as penalty if he has delayed the work by 30 days? (4 marks)
Q25. If the pth term of an AP is 1/q and qth term is 1/p, prove that the sum of first pq terms is (pq + 1)/2. (5 marks)

Case-Study Based Question (4 marks)

Case-study questions present a real-world scenario followed by 3-4 sub-questions testing your ability to extract mathematical information from context and apply AP concepts. Read the case carefully and identify all numerical data and relationships. Often, you need to recognize that the situation describes an AP even when not explicitly stated. Sub-questions progressively increase in difficulty. The first sub-question is usually direct, while later ones may require forming equations or combining formulas. These questions mirror the latest CBSE board exam pattern introduced from 2020-21 onwards and carry significant weightage.

Case Study: A school auditorium has seats arranged in rows. The first row has 20 seats, the second row has 23 seats, the third row has 26 seats, and so on.
(i) How many seats are there in the 15th row? (1 mark)
(ii) If there are total 25 rows, what is the total seating capacity? (1 mark)
(iii) In which row are there exactly 62 seats? (2 marks)
OR (iii) If the auditorium can accommodate maximum 1000 people, what is the maximum number of complete rows that can be filled? (2 marks)

Answer Key with Explanations

This comprehensive answer key provides correct answers with concise explanations for every question in the worksheet. Use it only after attempting all questions yourself to maximize learning. For MCQs and objective questions, the correct option or value is stated with brief reasoning. For short and long answers, step-wise solutions are provided showing formula application and calculations. If your answer differs, trace back through the steps to identify where the error occurred. Common mistakes are highlighted in some explanations. Understanding these solutions is more important than just tallying marks. Review incorrect answers thoroughly and re-attempt them after a day or two to ensure concept retention.

Section A Answers: Multiple Choice Questions

Q1. (a) 32. Using aₙ = a + (n-1)d with a=5, d=3, n=10: a₁₀ = 5 + 9(3) = 32. Q2. (a) 4. Given a₇ = 34 and d = 5, so a + 6(5) = 34 gives a = 4. Q3. (d) 610. Here a=1, d=3, n=20. S₂₀ = 20/2[2(1) + 19(3)] = 10[2 + 57] = 590. Wait, recalculating: S₂₀ = 10[2 + 57] = 590, not 610. Correct answer is (b) 590. Q4. (c) 18th. Here a=21, d=-3. Setting 21 + (n-1)(-3) = -30 gives n = 18. Q5. (a) 0. Given 7a₇ = 11a₁₁. Using aₙ = a + (n-1)d: 7[a+6d] = 11[a+10d], simplifying gives 4a + 68d = 0, so a = -17d. Then a₁₈ = a + 17d = -17d + 17d = 0. Q6. (c) 38. Using l = a + (n-1)d: 350 = 17 + (n-1)9, solving gives n = 38.

Answer 1: (a) 32
Answer 2: (a) 4
Answer 3: (b) 590
Answer 4: (c) 18th
Answer 5: (a) 0
Answer 6: (c) 38

Section B, C, D & E Answers with Working

Section B Answers: Q7. 1/6 (common difference between consecutive fractions). Q8. 40 (using sum formula with a=-5, d=2, n=10). Q9. 158 (20th term from end in an AP is same as 20th term of reversed AP; first find total terms then work backwards). Q10. 6 (since Sₙ = 3n² + 5n, find S₁ and S₂, then a₂ - a₁ gives d). Q11. 7 (if three numbers are a-d, a, a+d with sum 21, then 3a=21 so a=7). Q12. 16 (solving 7 + (n-1)3 = 52 gives n=16). Section C Answers: Q13. False (not every sequence has constant difference). Q14. True (d=0 means all terms identical). Q15. True (standard formula). Q16. True (decreasing AP has d<0). Q17. False (counter-example: 1,2,3 are in AP but 1,4,9 are not). Section D Answers: Q18. 178 (work shown in example above). Q19. 30 numbers (AP: 12,15,18,...,99 has 30 terms). Q20. First three terms are 2,5,8 (form equations from given conditions and solve). Q21. 65th term (find a₅₄ first, then solve for n when aₙ = a₅₄ + 132). Q22. 70336 (AP: 105,112,...,994; find n then sum). Section E Answers: Q23. nth term aₙ = Sₙ - Sₙ₋₁ = 4n + 3; it is AP with d=4; a₂₀=83. Q24. ₹23,250 (AP with a=200, d=50, n=30; calculate S₃₀). Q25. Derive a and d from given conditions, substitute in sum formula, simplify to get (pq+1)/2.

Section B: Answers are 1/6, 40, 158, 6, 7, 16 respectively
Section C: F, T, T, T, F respectively
Section D: Detailed step-by-step working required for full marks in exam
Section E: Q23 shows aₙ = 4n+3; Q24 penalty is ₹23,250; Q25 requires algebraic proof
Case Study: (i) 62 seats, (ii) 1325 seats, (iii) 15th row or maximum 19 complete rows depending on option chosen

How CBSETUTOR.ai Helps Master Arithmetic Progressions

Many Class 10 students struggle with word problems in Arithmetic Progressions or make algebraic errors while forming equations from given conditions. CBSETUTOR.ai provides a 24×7 AI tutor that students can access anytime they are stuck on a problem. Simply snap a photo of the question using your phone, upload it to the platform, and receive step-by-step solutions instantly. The AI tutor explains not just the answer but the reasoning behind choosing specific formulas and how to avoid common mistakes. For a flat fee of ₹999 per month covering all subjects for classes 6-12, students get unlimited doubt solving, practice worksheets, and personalized feedback. Parents appreciate the affordability compared to traditional coaching which can cost ₹5,000-15,000 monthly. The 3-day free trial allows students to test the platform with actual Class 10 Mathematics Chapter 5 problems before committing. This on-demand support is especially valuable during exam revision when doubts arise late at night or on weekends.

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Frequently asked questions

How many marks does Arithmetic Progressions carry in CBSE Class 10 board exam?+

Chapter 5 Arithmetic Progressions typically carries 6 marks in the CBSE Class 10 Mathematics board exam. This usually comprises one 2-mark short-answer question, one 3-mark question, and sometimes part of a 4-mark case-study question. The chapter is considered moderately scoring if formulas are applied correctly.

What are the most important formulas for nth term and sum of n terms?+

The two critical formulas are: (1) nth term aₙ = a + (n-1)d where a is first term, d is common difference, n is term number; (2) Sum of n terms Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2[a + l] where l is last term. Memorize both forms of the sum formula as different problems suit different forms.

How do I identify whether a given sequence is an Arithmetic Progression?+

Calculate the difference between consecutive terms. If the difference remains constant throughout, the sequence is an AP. For example, in 5,9,13,17, the differences are 4,4,4 (constant), so it is an AP. If differences vary like in 2,4,7,11, it is not an AP. Alternatively, check if aₙ₊₁ - aₙ is the same for all n.

What is the common mistake students make in word problems on Arithmetic Progressions?+

The most frequent error is incorrect identification of the first term 'a' and common difference 'd' from the problem statement. Students often confuse which quantity represents a and which represents d. Always list out at least the first three terms explicitly from the word problem before applying formulas. Also, carefully note whether the problem asks for the term number or the term value itself.

How should I approach case-study questions on Arithmetic Progressions?+

Read the entire scenario twice and underline all numerical data. Identify which quantities form the AP — often real-world contexts like seat arrangements, salary increments, or construction schedules. Extract the first term and common difference from the context, then treat sub-questions as standard AP problems using appropriate formulas. The first sub-question usually tests direct formula application while later ones require equation formation.

Is it necessary to learn the derivation of sum of n terms formula?+

While CBSE board exams rarely ask for derivation directly, understanding how Sₙ = n/2[2a + (n-1)d] is derived helps you remember the formula correctly and apply it confidently. The derivation using the method of writing the sum forwards and backwards is elegant and worth knowing. However, prioritize formula application practice over derivation memorization for board exam scoring.

How much time should I spend on this worksheet for effective practice?+

Allocate 90 minutes for the first attempt under timed conditions to simulate exam pressure. If you are practising for the first time, you may extend to 120 minutes. After completing, spend another 30-40 minutes reviewing the answer key and understanding mistakes. Re-attempt incorrect questions after 2-3 days. Completing 3-4 such full-length worksheets ensures thorough chapter mastery.

What are the easiest marks to score in Arithmetic Progressions chapter?+

MCQs and fill-in-the-blanks based on direct formula substitution are the easiest marks. Questions asking for the nth term when first term and common difference are directly given, or sum calculations with all values provided, require only careful arithmetic. Focus on accuracy in these sections first before attempting complex word problems. These objective questions typically constitute 3-4 marks and should be attempted with 100% accuracy target.

Can Arithmetic Progressions formulas be used for geometric sequences?+

No, AP formulas apply only when the difference between consecutive terms is constant. Geometric Progressions (GP) have a constant ratio between consecutive terms and require different formulas. Mixing these concepts is a common error. If a problem mentions percentage increase or multiplicative growth, it likely involves GP not AP. Always verify whether common difference or common ratio is constant before selecting formulas.

How does CBSETUTOR.ai help if I get stuck on a specific AP problem at night?+

CBSETUTOR.ai operates 24×7, so you can photograph your problem using your smartphone and upload it anytime — even at 11 PM while revising. The AI tutor provides step-by-step solutions within seconds, explaining which formula to use and why. This eliminates waiting until next day for teacher or tutor help. The platform costs just ₹999/month for unlimited access across all subjects and classes 6-12, with a 3-day free trial to test it first.

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