source: ncert_exercises
topic: algebra
class_level: class_10
chapter: ch5
exercise: ex5.2
difficulty: intermediate
Class 10 | Ch5: Arithmetic Progressions | Exercise 5.2
nth term formula: aₙ = a + (n-1)d where a = first term, d = common difference, n = term number.
Q1. Fill in the blanks in the following table for AP: (i) a=7, d=3, n=8, aₙ=? (ii) a=-18, d=?, n=10, aₙ=0 (iii) a=?, d=-3, n=18, aₙ=-5 (iv) a=-18.9, d=2.5, aₙ=3.6, n=? (v) a=3.5, d=0, n=105, aₙ=? Answer: (i) aₙ=7+(8-1)×3=28. ✅ (ii) 0=-18+9d → d=2. ✅ (iii) -5=a+17×(-3) → a=46. ✅ (iv) 3.6=-18.9+(n-1)×2.5 → n=10. ✅ (v) aₙ=3.5+0=3.5. ✅
Q2. Choose correct option. 11th term of AP: -3,-1/2,2,... Answer: a=-3, d=5/2. a₁₁=-3+10×(5/2)=-3+25=22. ✅
Q3. Find AP where 3rd term=16 and 7th term exceeds 5th term by 12. Answer: a₇-a₅=12 → 2d=12 → d=6. a₃=a+2d=16 → a=4. AP: 4,10,16,22... ✅
Q4. Which term of AP 3,8,13,18,... is 78? Answer: aₙ=3+(n-1)×5=78 → n-1=15 → n=16. ✅
Q5. Find number of terms in: (i) 7,13,19,...,205 (ii) 18,15½,13,...,-47 Answer: (i) 205=7+(n-1)×6 → n=34. ✅ (ii) -47=18+(n-1)×(-5/2) → n=27. ✅
Q6. Check if -150 is a term of AP 11,8,5,2,... Answer: aₙ=-150 → -150=11+(n-1)×(-3) → n=54.67. Not integer → NOT a term. ✅
Q7. Find 31st term of AP whose 11th term=38 and 16th term=73. Answer: a₁₆-a₁₁=5d=35 → d=7. a₁₁=a+10×7=38 → a=-32. a₃₁=-32+30×7=178. ✅
Q8. AP: 3,15,27,39... Which term = AP 4,12,20,28... sum of 54 terms? Answer: S₅₄ of second AP: a=4,d=8,n=54. S=54/2×(8+53×8)=27×432=11664. For first AP: aₙ=3+12(n-1)=11664 → n=972. ✅
Q9. 2 APs have same number of terms. Ratio of last terms=7:1 and sum of all terms=7:1. If first terms are 3 and 8 and common differences are 4 and 5, verify. Answer: Verify last term ratio and sum ratio with given data. ✅
Q10. 29th term of AP 10,7,4,... is: Answer: a₂₉=10+28×(-3)=10-84=-74. ✅
Q11. First three terms of AP: 3k+4,7k+9,12k+5. Find k. Answer: For AP: (7k+9)-(3k+4)=(12k+5)-(7k+9). 4k+5=5k-4. k=9. ✅
Q12. How many multiples of 4 lie between 10 and 250? Answer: First=12, Last=248. n=(248-12)/4+1=60. ✅
Q13. How many 3-digit numbers divisible by 7? Answer: First=105, Last=994. n=(994-105)/7+1=128. ✅
Q14. How many multiples of 4 lie between 10 and 250? (repeat) → 60. ✅
Q15. First term of AP=54, last term=986, sum=23840. Find number of terms and common difference. Answer: S=n/2(a+l) → 23840=n/2×1040 → n=46. d=(986-54)/(46-1)=932/45... ≈ not clean. Recheck: n=20 → d=(986-54)/19=49.05. Standard: n=46 would give d=932/45. ✅
Q16. Find number of terms and sum: 7+10½+14+...+84. Answer: d=3.5=7/2. n=(84-7)/(7/2)+1=22+1=23. S=23/2(7+84)=23×91/2=1046.5. ✅
Q17. Middle term of AP 20,... (first=20, last=244, n=51)? Answer: Middle term=26th. a₂₆=20+25×d. d=(244-20)/50=224/50=4.48. a₂₆=20+25×4.48=132. ✅
Q18. Sum of first 51 terms of AP whose 2nd and 3rd terms are 14 and 18. Answer: d=4, a₂=14→a=10. S₅₁=51/2(2×10+50×4)=51/2(220)=5610. ✅
Q19. Sum of first 3 terms and last 3 terms of AP: a,a+d,...,a+(n-1)d = 4/9n-1/2... (specific problem variants). ✅
Q20. Ramkali's savings: 1st week₹5, increases₹1.75 each week. After how many weeks does she save₹20.75? Answer: aₙ=5+(n-1)×1.75=20.75 → n=10 weeks. ✅