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).

Key Themes & Tips

• Sequences: Constant difference.
• Formulas: nth, sum.
• Applications: Real patterns.
• Tip: Check d same; use formulas directly.

Project & Group Ideas

• Model patterns; derive sum formula.

Key Definitions & Terms - Complete Glossary

All terms from chapter; easy to memorize.

Tip: Use flashcards; remember formulas with examples.

60+ Questions & Answers - NCERT Based (Class 10)

Structured as Part A (1 mark, short answers), Part B (4 marks, ~6 lines answers), Part C (8 marks, detailed). 20 per part, based on chapter content, with answers matching the mark scheme.

Part A: 1 Mark Questions (Short Answers)

Part B: 4 Marks Questions (Answers in ~6 Lines)

Part C: 8 Marks Questions (Detailed Answers)

Practice Tip: Time yourself; use formulas for long Q.

Important Formulas

Key equations for solving/revision.

Use in: Problems for quick solve.

Theorems & Proofs - Detailed Explanations

Key theorems with step-by-step proofs and examples.

Tip: Use for verifying sums.

Derivations - Step by Step

• nth Term Derivation: Start a1=a. a2=a1+d=a+d. a3=a2+d=a+2d. a4=a+3d. Pattern an=a+(n-1)d.
• Sum Derivation: S = a + (a+d) + ... + l. Reverse S = l + (l-d) + ... + a. Add 2S = n(a+l). S=n/2 (a+l). Since l=a+(n-1)d, S=n/2 [2a+(n-1)d].

Tip: Practice for understanding.

All Textbook Examples - Step by Step Solutions

Complete list with detailed explanations like the book, step by step.

Tip: Practice variations.

Methods of Solving - Step by Step

• Identify AP: Check ak+1 - ak same. Example: Differences constant.
• Find nth Term: an = a + (n-1)d. Solve for n if an known.
• Find Sum: Sn = n/2 [2a + (n-1)d]. Or n/2 (a+l).
• Number of Terms: From an = l, n = (l - a)/d +1.
• From Two Terms: Solve system for a,d.

Choose: Formulas direct; check integer n.

Applications & Word Problems - Real Life

• Salary Increments: Annual raise. Example: 8000 +500 yearly.
• Ladder Rungs: Decreasing lengths. Example: 45 to 31 cm.
• Savings Schemes: Compound but if simple AP. Example: Interest AP.
• Prizes/Penalties: Increasing. Example: Penalty 200+50 daily.
• Stacked Items: Logs decreasing. Example: 20 bottom to 1.
• Race Distances: Potato back-forth. Example: 5m +3m apart.

Tip: Form AP, use formulas.

Quick Revision Notes & Mnemonics - Exam-Ready

Mnemonics: AP: Add Permanently (constant d). Sum: Number Half First Last (n/2 (a+l)).

Tip: Check d; positive n.