Complete Solutions and Summary of Arithmetic Progressions – NCERT Class 10, Mathematics, Chapter 5 – Summary, Questions, Answers, Extra Questions
Comprehensive summary and explanation of Chapter 5 'Arithmetic Progressions', covering the definition and identification of APs, derivation of the general term, finding the nth term, calculation of the sum of n terms, practical examples, and a variety of solved and extra practice problems drawn from daily life and real-world scenarios, all aligned with NCERT Class X Mathematics.
Updated: 8 months ago
Arithmetic Progressions
Chapter 5: Mathematics - Ultimate Study Guide | NCERT Class 10 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Arithmetic Progressions Class 10 NCERT
Overview & Key Concepts
- Chapter Goal: Understand arithmetic progressions, nth term, sum of terms, applications. Exam Focus: Identifying AP, formulas, word problems. 2025 Updates: Real-life patterns. Fun Fact: Gauss summed 1 to 100 quickly. Core Idea: Sequences with constant difference. Real-World: Salary increments, ladder rungs, savings.
- Wider Scope: Foundation for sequences, series in higher maths.
5.1 Introduction
- Patterns in nature: sunflower petals, honeycomb, maize grains, pineapple spirals.
- Daily life examples: Salary 8000 +500 annual, rungs 45,43,...,31 cm, savings 5/4 times every 3 years, unit squares 1²,2²,..., money box 100+50 yearly, rabbits 1,1,2,3,5,8 (Fibonacci, not AP).
- Patterns: Adding fixed, multiplying, squares, etc. Focus on adding fixed (AP).
5.2 Arithmetic Progressions
- List where each term = previous + fixed d (common difference).
- Examples: 1,2,3,... (d=1); 100,70,40,10 (d=-30); -3,-2,-1,0 (d=1); 3,3,3 (d=0); -1.0,-1.5,-2.0 (d=-0.5).
- General: a, a+d, a+2d,...
- Finite (last term), infinite (no last).
- Check AP: Differences same.
Example 1: First Term and d
- For 3/2, 1/2, -1/2, -3/2,... a=3/2, d=-1.
Example 2: Check AP and Next Terms
- (i) 4,10,16,22,... d=6, yes; next 28,34.
- (ii) 1,-1,-3,-5,... d=-2, yes; next -7,-9.
- (iii) -2,2,-2,2,... differences 4,-4, not same, no.
- (iv) 1,1,1,2,2,2,3,... differences 0,0,1, not same, no.
Exercise 5.1
- Situations form AP? Taxi fare, air removal, well digging, compound interest.
- First four terms given a,d.
- First term, d from APs.
- Which are APs? Write next three.
5.3 nth Term of an AP
- an = a + (n-1)d
- Last term l = a + (m-1)d for m terms.
Example 3: 10th Term
- 2,7,12,... a=2,d=5; a10=2+9*5=47.
Example 4: Which Term -81
- 21,18,15,... a=21,d=-3; 21+(n-1)(-3)=-81 → n=35. For 0: n=8.
Example 5: Determine AP
- a3=5, a7=9; solve a+2d=5, a+6d=9 → d=1, a=3. AP:3,4,5,6,...
Example 6: Check 301 in 5,11,17,23,...
- a=5,d=6; 5+(n-1)6=301 → n=302/6=50.333, not integer, no.
Example 7: Two-Digit Divisible by 3
- 12,15,...,99; a=12,d=3; 12+(n-1)3=99 → n=30.
Example 8: 11th from Last
- 10,7,4,...,-62; a=10,d=-3; n=25 terms. 11th from last =15th from first:10+14*(-3)=-32. Reverse: a=-62,d=3; a11=-62+10*3=-32.
Example 9: Interest 1000 at 8%
- Interests 80,160,240,... d=80,a=80. a30=80+29*80=2400.
Example 10: Rose Plants
- 23,21,19,...,5; a=23,d=-2; 23+(n-1)(-2)=5 → n=10.
Exercise 5.2
- Fill table a,d,n,an.
- Choose correct 30th/11th term.
- Missing terms in boxes.
- Which term 78.
- Number of terms in APs.
- Check term.
- 31st term given a11,a16.
- 50 terms, a3=12, last=106, find a29.
- Term zero.
- Common d exceeds by 7.
- Term 132 more.
- Same d, difference 100th=100, 1000th?
- Three-digit by 7.
- Multiples 4 between 10-250.
- Value n equal terms two APs.
- Determine AP a3=16, a7-a5=12.
- 20th from last 3,8,...,253.
- Sum a4+a8=24, a6+a10=44, first three.
- Salary 5000 +200 yearly, year 7000.
- Savings 5 +1.75 weekly, nth=20.75.
5.4 Sum of First n Terms of an AP
- Sn = n/2 [2a + (n-1)d] or n/2 (a + l)
- Gauss sum 1 to 100=5050.
- an = Sn - Sn-1
Example 11: Sum 22 Terms
- 8,3,-2,... a=8,d=-5; S22=22/2 [2*8 +21*(-5)]=11*(16-105)=-979.
Example 12: Sum 14 Terms=1050, a=10, Find a20
- S14=14/2 [2*10 +13 d]=1050 → 140+91d=1050 → d=10. a20=10+19*10=200.
Example 13: Terms Sum 78
- 24,21,18,... Sn=78. a=24,d=-3; n/2 [48 + (n-1)(-3)]=78 → n(51-3n)=156 → 3n²-51n+156=0 → n²-17n+52=0 → (n-4)(n-13)=0. n=4 or 13.
Example 14: Sum Positive Integers
- (i) 1 to 1000: 1000/2 (1+1000)=500500.
- (ii) 1 to n: n/2 (n+1).
Example 15: Sum 24 Terms an=3+2n
- a1=5,a2=7,... d=2. S24=24/2 [2*5 +23*2]=12*(10+46)=672.
Example 16: TV Production
- a3=600, a7=700; a+2d=600, a+6d=700 → d=25, a=550. a10=550+9*25=775. S7=7/2 [2*550 +6*25]=7/2*1250=4375.
Exercise 5.3
- Sum APs: 2,7,...10 terms; -37,-33,...12; 0.6,1.7,...100; 1/15,1/12,...11.
- Sums: 7+10.5+...+84; 34+32+...+10; -5-8-...-230.
- AP given values find others.
- Terms sum 636.
- a=5,l=45,S=400 find n,d.
- a=17,l=350,d=9 find n,S.
- Sum 22 terms d=7 a22=149.
- Sum 51 terms a2=14 a3=18.
- S7=49 S17=289 find Sn.
- Show an=3+4n or 9-5n, S15.
- Sn=4n-n² find a1,S2,a2,a3,a10,an.
- Sum 40 multiples 6.
- Sum 15 multiples 8.
- Sum odds 0-50.
- Penalty 200+50 daily, 30 days.
- Prizes 700 total, 20 less each.
- Trees sections classes 1-12, 3 sections.
- Semicircles length 13.
- Logs 200 stacked 20 bottom decreasing.
- Potato race distance 10 potatoes.
5.5 Summary
- AP: Each term +d to previous.
- ak+1 - ak =d same.
- an =a+(n-1)d.
- Sn =n/2 [2a+(n-1)d] or n/2 (a+l).
Why This Guide Stands Out
Complete chapter coverage: Notes, examples, Q&A (all NCERT + extras), quiz. Student-centric, exam-ready for 2025. Free & ad-free.
Key Themes & Tips
- Sequences: Constant difference.
- Formulas: nth, sum.
- Applications: Real patterns.
- Tip: Check d same; use formulas directly.
Exam Case Studies
Word problems on sums, terms, applications like logs, prizes.
Project & Group Ideas
- Model patterns; derive sum formula.
As an Amazon Associate, ProSyllabus earns from qualifying purchases. Prices shown are subject to change.
Test your CBSE Class 10 Board Examination prep
Quizzes
10 questions · ~10 minutes · instant rank & AI diagnosis
Class 10 English — Nelson Mandela: Long Walk to Freedom (Practice Quiz)
Chemical Reactions and Equations Fundamentals — Free CBSE Class 10 Board Examination Quiz
Class 10 English — A Triumph of Surgery (Practice Quiz)
Class 10 English — A Letter to God (Practice Quiz)
Reading Comprehension Passages Fundamentals | CBSE Class 10 Board Examination
कक्षा 10 संस्कृत — रचना-प्रयोगः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — वाच्य-परिवर्तनम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — कारक एवं विभक्ति (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — समास परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — प्रत्ययाः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — अव्ययानि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — उपसर्गाः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — धातुरूप परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — शब्दरूप परिचय (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सन्धिः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — संज्ञा एवं परिभाषा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — वर्णविचारः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — भूकंपविभीषिका (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सूक्तयः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — विचित्रः साक्षी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सौहार्दं प्रकृतेः शोभा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — सुभाषितानि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — जननी तुल्यवत्सला (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — शिशुलालनम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — व्यायामः सर्वदा पथ्यः (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — बुद्धिर्बलवती सदा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 संस्कृत — शुचिपर्यावरणम् (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — टोपी शुक्ला (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — सपनों के-से दिन (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — हरिहर काका (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — पतझर में टूटी पत्तियाँ, कारतूस (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — अब कहाँ दूसरे के दुख से दुखी होने वाले (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — गिरगिट (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तीसरी कसम के शिल्पकार शैलेंद्र (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तताँरा-वामीरो कथा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — डायरी का एक पन्ना (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बड़े भाई साहब (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — तोप, कर चले हम फ़िदा, आत्मत्राण (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मधुर-मधुर मेरे दीपक जल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — पर्वत प्रदेश में पावस (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मनुष्यता (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बिहारी के दोहे (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मीरा के पद (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — कबीर की साखी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — मैं क्यों लिखता हूँ (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — साना-साना हाथ जोड़ि (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — माता का अँचल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — संस्कृति (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — नौबतखाने में इबादत (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — एक कहानी यह भी (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — लखनवी अंदाज़ (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — बालगोबिन भगत (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — नेताजी का चश्मा (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — संगतकार (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — यह दंतुरित मुसकान, फसल (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — उत्साह, अट नहीं रही है (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — आत्मकथ्य (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — राम-लक्ष्मण-परशुराम संवाद (अभ्यास प्रश्नोत्तरी)
कक्षा 10 हिंदी — सूरदास के पद (अभ्यास प्रश्नोत्तरी)
Class 10 English — For Anne Gregory (Practice Quiz)
Class 10 English — The Tale of Custard the Dragon (Practice Quiz)
Class 10 English — Fog (Practice Quiz)
Class 10 English — The Trees (Practice Quiz)
Class 10 English — Amanda! (Practice Quiz)
Class 10 English — The Ball Poem (Practice Quiz)
Class 10 English — How to Tell Wild Animals (Practice Quiz)
Class 10 English — A Tiger in the Zoo (Practice Quiz)
Class 10 English — Fire and Ice (Practice Quiz)
Class 10 English — Dust of Snow (Practice Quiz)
Class 10 Health & PE — Doping and Ethics in Sports (Practice Quiz)
Class 10 Health & PE — Adolescent Health and Wellbeing (Practice Quiz)
Class 10 Health & PE — Diseases: Communicable and Non-communicable (Practice Quiz)
Class 10 Health & PE — The Olympic Movement (Practice Quiz)
Class 10 Health & PE — Common Sports Injuries and Prevention (Practice Quiz)
Class 10 Health & PE — First Aid and Safety Education (Practice Quiz)
Class 10 Health & PE — Yoga and Healthy Living (Practice Quiz)
Class 10 Health & PE — Posture and Postural Deformities (Practice Quiz)
Class 10 Health & PE — Food and Nutrition for Health (Practice Quiz)
Class 10 Health & PE — Effects of Exercise on the Body Systems (Practice Quiz)
Class 10 Health & PE — Physical Fitness, Wellness and Lifestyle (Practice Quiz)
Class 10 Health & PE — Physical Education: Meaning and Importance (Practice Quiz)
Class 10 English — The Book That Saved the Earth (Practice Quiz)
Class 10 English — Bholi (Practice Quiz)
Class 10 English — The Necklace (Practice Quiz)
Class 10 English — The Making of a Scientist (Practice Quiz)
Class 10 English — Footprints Without Feet (Practice Quiz)
Class 10 English — A Question of Trust (Practice Quiz)
Class 10 English — The Midnight Visitor (Practice Quiz)
Class 10 English — The Thief’s Story (Practice Quiz)
Class 10 English — The Proposal (Practice Quiz)
Class 10 English — The Sermon at Benares (Practice Quiz)
Class 10 English — Madam Rides the Bus (Practice Quiz)
Class 10 English — Mijbil the Otter (Practice Quiz)
Class 10 English — Glimpses of India (Practice Quiz)
Class 10 English — From the Diary of Anne Frank (Practice Quiz)
Class 10 English — Two Stories about Flying (Practice Quiz)
Class 10 Social Science — Consumer Rights (Practice Quiz)
Class 10 Social Science — Globalisation and the Indian Economy (Practice Quiz)
Class 10 Social Science — Money and Credit (Practice Quiz)
Class 10 Social Science — Sectors of the Indian Economy (Practice Quiz)
Class 10 Social Science — Development (Practice Quiz)
Class 10 Social Science — Outcomes of Democracy (Practice Quiz)
Class 10 Social Science — Political Parties (Practice Quiz)
Class 10 Social Science — Gender, Religion and Caste (Practice Quiz)
Class 10 Social Science — Federalism (Practice Quiz)
Class 10 Social Science — Power-sharing (Practice Quiz)
Class 10 Social Science — Lifelines of National Economy (Practice Quiz)
Class 10 Social Science — Manufacturing Industries (Practice Quiz)
Class 10 Social Science — Minerals and Energy Resources (Practice Quiz)
Class 10 Social Science — Agriculture (Practice Quiz)
Class 10 Social Science — Water Resources (Practice Quiz)
Class 10 Social Science — Forest and Wildlife Resources (Practice Quiz)
Class 10 Social Science — Resources and Development (Practice Quiz)
Class 10 Social Science — Print Culture and the Modern World (Practice Quiz)
Class 10 Social Science — The Age of Industrialisation (Practice Quiz)
Class 10 Social Science — The Making of a Global World (Practice Quiz)
Class 10 Social Science — Nationalism in India (Practice Quiz)
Class 10 Social Science — The Rise of Nationalism in Europe (Practice Quiz)
Class 10 Science — Our Environment (Practice Quiz)
Class 10 Science — Magnetic Effects of Electric Current (Practice Quiz)
Class 10 Science — Electricity (Practice Quiz)
Class 10 Science — The Human Eye and the Colourful World (Practice Quiz)
Class 10 Science — Light – Reflection and Refraction (Practice Quiz)
Class 10 Science — Heredity (Practice Quiz)
Class 10 Science — How do Organisms Reproduce? (Practice Quiz)
Class 10 Science — Control and Coordination (Practice Quiz)
Class 10 Science — Life Processes (Practice Quiz)
Class 10 Science — Carbon and its Compounds (Practice Quiz)
Class 10 Science — Metals and Non-metals (Practice Quiz)
Class 10 Science — Acids, Bases and Salts (Practice Quiz)
Class 10 Maths — Probability (Practice Quiz)
Class 10 Maths — Statistics (Practice Quiz)
Class 10 Maths — Surface Areas and Volumes (Practice Quiz)
Class 10 Maths — Areas Related to Circles (Practice Quiz)
Class 10 Maths — Circles (Practice Quiz)
Class 10 Maths — Some Applications of Trigonometry (Practice Quiz)
Class 10 Maths — Introduction to Trigonometry (Practice Quiz)
Class 10 Maths — Coordinate Geometry (Practice Quiz)
Class 10 Maths — Triangles (Practice Quiz)
Class 10 Maths — Arithmetic Progressions (Practice Quiz)
Class 10 Maths — Quadratic Equations (Practice Quiz)
Class 10 Maths — Pair of Linear Equations in Two Variables (Practice Quiz)
Class 10 Maths — Polynomials (Practice Quiz)
Class 10 Maths — Real Numbers (Practice Quiz)
Class 10 Science — Chemical Reactions and Equations (Practice Quiz)
Political Science: Democratic Politics II Practice Quiz | CBSE Class 10 Board Examination
Economics: Understanding Economic Development Practice Quiz | CBSE Class 10 Board Examination
Hindi A/B (Kshitij II / Sparsh II / Kritika II / Sanchayan II) Practice Quiz | CBSE Class 10 Board Examination
Carbon and Its Compounds Advanced Challenge | CBSE Class 10 Board Examination
Group Discussions
No forum posts available.
