April 10, 2026
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Chapter 5: Arithmetic Progressions
Complete NCERT Solutions | CBSE Class 10 Mathematics (2025-26)
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Exercise 5.1 (Page 55)
12 marks each
In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?
(i) Taxi fare: ₹15 for first km, ₹8 for each additional km.
(ii) Air in cylinder: vacuum pump removes 1/4 of remaining air each time.
(iii) Cost of digging well: ₹150 for first metre, rises by ₹50 each subsequent metre.
(iv) Money in account: ₹10000 deposited at 8% compound interest per annum.
(i) Taxi fare: ₹15 for first km, ₹8 for each additional km.
(ii) Air in cylinder: vacuum pump removes 1/4 of remaining air each time.
(iii) Cost of digging well: ₹150 for first metre, rises by ₹50 each subsequent metre.
(iv) Money in account: ₹10000 deposited at 8% compound interest per annum.
Answer:
(i) Fares: 15, 23, 31, 39, ... → Difference constant = 8 → Yes, AP
(ii) Air left: V, 3V/4, 9V/16, ... → Ratio constant, not difference → Not AP
(iii) Costs: 150, 200, 250, ... → Difference constant = 50 → Yes, AP
(iv) Compound interest: Amount grows by multiplying factor, not constant difference → Not AP
(i) Fares: 15, 23, 31, 39, ... → Difference constant = 8 → Yes, AP
(ii) Air left: V, 3V/4, 9V/16, ... → Ratio constant, not difference → Not AP
(iii) Costs: 150, 200, 250, ... → Difference constant = 50 → Yes, AP
(iv) Compound interest: Amount grows by multiplying factor, not constant difference → Not AP
21 mark each
Write first four terms of AP when (i) a=10, d=10 (ii) a=-2, d=0 (iii) a=4, d=-3 (iv) a=-1, d=1/2 (v) a=-1.25, d=-0.25
Answer:
(i) 10, 20, 30, 40
(ii) -2, -2, -2, -2
(iii) 4, 1, -2, -5
(iv) -1, -0.5, 0, 0.5
(v) -1.25, -1.50, -1.75, -2.00
(i) 10, 20, 30, 40
(ii) -2, -2, -2, -2
(iii) 4, 1, -2, -5
(iv) -1, -0.5, 0, 0.5
(v) -1.25, -1.50, -1.75, -2.00
31 mark each
For the following APs, write first term and common difference:
(i) 3, 1, -1, -3, ... (ii) -5, -1, 3, 7, ... (iii) 1/3, 5/3, 9/3, 13/3, ... (iv) 0.6, 1.7, 2.8, 3.9, ...
(i) 3, 1, -1, -3, ... (ii) -5, -1, 3, 7, ... (iii) 1/3, 5/3, 9/3, 13/3, ... (iv) 0.6, 1.7, 2.8, 3.9, ...
Answer:
(i) a=3, d=-2
(ii) a=-5, d=4
(iii) a=1/3, d=4/3
(iv) a=0.6, d=1.1
(i) a=3, d=-2
(ii) a=-5, d=4
(iii) a=1/3, d=4/3
(iv) a=0.6, d=1.1
41 mark each
Which of the following are APs? If they form an AP, find common difference d and write three more terms.
(i) 2,4,8,16,... (ii) 2,5/2,3,7/2,... (iii) -1.2,-3.2,-5.2,-7.2,... (iv) -10,-6,-2,2,... (v) 3,3+√2,3+2√2,3+3√2,... (vi) 0.2,0.22,0.222,0.2222,... (vii) 0,-4,-8,-12,... (viii) -1/2,-1/2,-1/2,-1/2,... (ix) 1,3,9,27,... (x) a,2a,3a,4a,... (xi) a,a²,a³,a⁴,... (xii) √2,√8,√18,√32,... (xiii) √3,√6,√9,√12,... (xiv) 1²,3²,5²,7²,... (xv) 1²,5²,7²,73,...
(i) 2,4,8,16,... (ii) 2,5/2,3,7/2,... (iii) -1.2,-3.2,-5.2,-7.2,... (iv) -10,-6,-2,2,... (v) 3,3+√2,3+2√2,3+3√2,... (vi) 0.2,0.22,0.222,0.2222,... (vii) 0,-4,-8,-12,... (viii) -1/2,-1/2,-1/2,-1/2,... (ix) 1,3,9,27,... (x) a,2a,3a,4a,... (xi) a,a²,a³,a⁴,... (xii) √2,√8,√18,√32,... (xiii) √3,√6,√9,√12,... (xiv) 1²,3²,5²,7²,... (xv) 1²,5²,7²,73,...
Answer:
(i) Not AP (ratio constant)
(ii) AP, d=0.5, next: 4, 9/2, 5
(iii) AP, d=-2, next: -9.2, -11.2, -13.2
(iv) AP, d=4, next: 6, 10, 14
(v) AP, d=√2, next: 3+4√2, 3+5√2, 3+6√2
(vi) Not AP (differences not constant)
(vii) AP, d=-4, next: -16, -20, -24
(viii) AP, d=0, next: -1/2, -1/2, -1/2
(ix) Not AP (ratio constant)
(x) AP, d=a, next: 5a, 6a, 7a
(xi) Not AP (ratio constant)
(xii) AP, d=√2, next: √50, √72, √98
(xiii) Not AP (√3,√6,√9 differences not equal)
(xiv) AP, d=8, next: 9²=81, 11²=121, 13²=169
(xv) Not AP (numbers not following pattern)
(i) Not AP (ratio constant)
(ii) AP, d=0.5, next: 4, 9/2, 5
(iii) AP, d=-2, next: -9.2, -11.2, -13.2
(iv) AP, d=4, next: 6, 10, 14
(v) AP, d=√2, next: 3+4√2, 3+5√2, 3+6√2
(vi) Not AP (differences not constant)
(vii) AP, d=-4, next: -16, -20, -24
(viii) AP, d=0, next: -1/2, -1/2, -1/2
(ix) Not AP (ratio constant)
(x) AP, d=a, next: 5a, 6a, 7a
(xi) Not AP (ratio constant)
(xii) AP, d=√2, next: √50, √72, √98
(xiii) Not AP (√3,√6,√9 differences not equal)
(xiv) AP, d=8, next: 9²=81, 11²=121, 13²=169
(xv) Not AP (numbers not following pattern)
Exercise 5.2 (Page 61) – nth Term
11 mark each
Fill in blanks: (i) a=7,d=3,n=8 → a₈=? (ii) a=-18,d=?,n=10,aₙ=0 (iii) a=?,d=-3,n=18,aₙ=-5 (iv) a=-18.9,d=2.5,n=?,aₙ=3.6 (v) a=3.5,d=0,n=105,aₙ=?
Answer:
(i) a₈=7+7×3=28
(ii) d= (0+18)/9 = 2
(iii) a = -5 + 51 = 46
(iv) n = (3.6+18.9)/2.5 +1 = 10
(v) a₁₀₅ = 3.5
(i) a₈=7+7×3=28
(ii) d= (0+18)/9 = 2
(iii) a = -5 + 51 = 46
(iv) n = (3.6+18.9)/2.5 +1 = 10
(v) a₁₀₅ = 3.5
21 mark each
Choose correct: (i) 30th term of AP 10,7,4,... is (A)97 (B)77 (C)-77 (D)-87
(ii) 11th term of AP -3,-1/2,2,... is (A)28 (B)22 (C)-38 (D)-48
(ii) 11th term of AP -3,-1/2,2,... is (A)28 (B)22 (C)-38 (D)-48
Answer:
(i) d=-3, a₃₀ = 10+29×(-3) = -77 → (C)
(ii) d=2.5, a₁₁ = -3+10×2.5 = 22 → (B)
(i) d=-3, a₃₀ = 10+29×(-3) = -77 → (C)
(ii) d=2.5, a₁₁ = -3+10×2.5 = 22 → (B)
3-20Selected Solutions
Key results from remaining problems in Exercise 5.2:
4. 78 is 16th term of AP 3,8,13,...
5. (i) 34 terms in 7,13,...,205 (ii) 27 terms in 18,15.5,...,-47
6. -150 is not a term (n not integer)
7. a₃₁ = 178
8. a₂₉ = 64
9. 5th term is zero
10. d = 7
11. 54th term + 132 = 69th term
12. Difference remains 100
13. 128 three-digit numbers divisible by 7
14. 60 multiples of 4 between 10 and 250
15. n = 13
16. AP: 4, 10, 16, ...
17. 20th term from last = 158
18. First three terms: -13, -8, -3
19. Year 2005 (11th year)
20. n = 10
5. (i) 34 terms in 7,13,...,205 (ii) 27 terms in 18,15.5,...,-47
6. -150 is not a term (n not integer)
7. a₃₁ = 178
8. a₂₉ = 64
9. 5th term is zero
10. d = 7
11. 54th term + 132 = 69th term
12. Difference remains 100
13. 128 three-digit numbers divisible by 7
14. 60 multiples of 4 between 10 and 250
15. n = 13
16. AP: 4, 10, 16, ...
17. 20th term from last = 158
18. First three terms: -13, -8, -3
19. Year 2005 (11th year)
20. n = 10
Exercise 5.3 (Page 68) – Sum of n Terms
12 marks each
Find sum: (i) 2,7,12,... to 10 terms (ii) -37,-33,-29,... to 12 terms (iii) 0.6,1.7,2.8,... to 100 terms (iv) 1/15,1/12,1/10,... to 11 terms
Answer:
(i) S₁₀ = 5×(4+9×5)=245
(ii) S₁₂ = 6×(-74+11×4)= -180
(iii) S₁₀₀ = 50×(1.2+99×1.1)= 5505
(iv) S₁₁ = 11/2×(2/15+10×1/60)= 33/20 = 1.65
(i) S₁₀ = 5×(4+9×5)=245
(ii) S₁₂ = 6×(-74+11×4)= -180
(iii) S₁₀₀ = 50×(1.2+99×1.1)= 5505
(iv) S₁₁ = 11/2×(2/15+10×1/60)= 33/20 = 1.65
23 marks each
Find sums: (i) 7+10½+14+...+84 (ii) 34+32+30+...+10 (iii) -5+(-8)+(-11)+...+(-230)
Answer:
(i) S₂₃ = 23/2×(7+84)= 1046.5
(ii) S₁₃ = 13/2×(34+10)= 286
(iii) S₇₆ = 76/2×(-5-230)= -8930
(i) S₂₃ = 23/2×(7+84)= 1046.5
(ii) S₁₃ = 13/2×(34+10)= 286
(iii) S₇₆ = 76/2×(-5-230)= -8930
3-20Key Results
Selected answers from Exercise 5.3 problems 3-20:
3. (i) n=16, S=440 (ii) d=14/3, S=273 (iii) a=4, S=246 (iv) d=1, a₁₀=24 (v) a= -25/3, a₉= 35/3
(vi) n=5, aₙ=34 (vii) n=7, d=9 (viii) n=7, a= -8 (ix) d=6 (x) a=4
4. n=12 or 14 (both possible)
5. n=16, d=8/3
6. n=38, S=6973
7. S₂₂ = 1661
8. S₅₁ = 5610
9. Sₙ = n²
10. (i) S₁₅=555 (ii) S₁₅= -465
11. a₁=3, a₂=1, a₃= -1, a₁₀= -15, aₙ= 5-2n
12. S₄₀ = 4920
13. S₁₅ = 960
14. S₂₅ = 625 (odd numbers 1 to 49)
15. Penalty = ₹ 277500
16. Prizes: ₹160,140,120,100,80,60,40
17. Trees planted = 234
18. Total length = 143 cm
19. 16 rows, top row has 5 logs
20. Total distance = 370 m
(vi) n=5, aₙ=34 (vii) n=7, d=9 (viii) n=7, a= -8 (ix) d=6 (x) a=4
4. n=12 or 14 (both possible)
5. n=16, d=8/3
6. n=38, S=6973
7. S₂₂ = 1661
8. S₅₁ = 5610
9. Sₙ = n²
10. (i) S₁₅=555 (ii) S₁₅= -465
11. a₁=3, a₂=1, a₃= -1, a₁₀= -15, aₙ= 5-2n
12. S₄₀ = 4920
13. S₁₅ = 960
14. S₂₅ = 625 (odd numbers 1 to 49)
15. Penalty = ₹ 277500
16. Prizes: ₹160,140,120,100,80,60,40
17. Trees planted = 234
18. Total length = 143 cm
19. 16 rows, top row has 5 logs
20. Total distance = 370 m
Exercise 5.4 (Optional – Page 71)
1-5Challenge Problems
Selected answers from optional exercise:
1. 32nd term is first negative
2. S₁₆ = 76 or 20
3. Wood required = 3850 cm
4. x = 35
5. Volume of concrete = 750 m³
2. S₁₆ = 76 or 20
3. Wood required = 3850 cm
4. x = 35
5. Volume of concrete = 750 m³
Key Concepts Summary
Note 1General Form
What is an Arithmetic Progression (AP)?
Answer: AP is a list of numbers where each term is obtained by adding a fixed number (common difference d) to the preceding term. General form: a, a+d, a+2d, ...
Note 2nth Term Formula
What is the formula for the nth term of an AP?
Answer: aₙ = a + (n-1)d, where a = first term, d = common difference.
Note 3Sum of n Terms
What are the formulas for sum of first n terms of an AP?
Answer: Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2(a + l), where l = last term.
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