Class 10 Maths – Chapter 5: Arithmetic Progressions

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

Write first four terms of AP when: a=10, d=10 â†' 10,20,30,40. a=âˆ'2, d=0 â†' âˆ'2,âˆ'2,âˆ'2,âˆ'2.

Q3: Check which are APs: (i) 2,4,8,16 â†' not AP (geometric). (ii) 2,5/2,3,7/2 â†' d=1/2 â†' AP. (iii) âˆ'1.2,âˆ'3.2,âˆ'5.2,âˆ'7.2 â†' d=âˆ'2 â†' AP.

Q1: Find nth term of 2,4,6,8... â†' a=2, d=2. aâ‚™=2+2(nâˆ'1)=2n, a₁₅=30

Q2: Which term of 3,6,9,12... is 99? 3n=99 â†' n=33

Q3: Which term of 15,12,9,6... is âˆ'27? a=15, d=âˆ'3. 15+(nâˆ'1)(âˆ'3)=âˆ'27 â†' n=15

Q4: Which term of 4,9,14,19... is 109? â†' a=4, d=5. 4+5(nâˆ'1)=109 â†' n=22

Q5: Find missing terms in AP: 2, __, 26 â†' a=2, a₃=26. 2+2d=26 â†' d=12. Missing:14

Q6: How many three-digit numbers are divisible by 7? First=105, last=994. 105+(nâˆ'1)7=994 â†' n=128

Q7: Find 11th term from last of 10,7,4... âˆ'62. Last is âˆ'62. Reverse AP: a=âˆ'62, d=3. a₁₁=âˆ'62+10(3)=âˆ'32

Q11: Which term of AP 3, 15, 27... is 132 more than its 54th term? aâ‚…â‚„=3+53×12=639. aâ‚™=639+132=771. n=((771âˆ'3)/12)+1=65

Q16: 7th term is 4 times 2nd term. 12th term is 2 more than 3 times the 4th term. Find AP. â†' a=2, d=3. AP: 2,5,8,11...

Q17: 17th term exceeds 10th term by 7. Find common difference. â†' a₁₇âˆ'a₁₀=7d=7 â†' d=1

Q20: Ramkali saved ₹5 on first day, ₹7 on second, ₹9 on third... How many days to save ₹715? a=5, d=2. Sâ‚™=n/2[10+2(nâˆ'1)]=715 â†' n²+4nâˆ'715=0 â†' 25 days

Q1: Find sum of 2,7,12... to 10 terms. a=2, d=5. S₁₀=5[4+9×5]=5×49=245

Q2: Sum of âˆ'37,âˆ'33,âˆ'29... to 12 terms. S₁₂=6[âˆ'74+11×4]=6×(âˆ'30)=âˆ'180

Q3: Given aₙ=3n+2, find sum of first 10 terms. a=5, a₁₀=32. S₁₀=5(5+32)=185

Q6: Sum of first 20 terms of AP with a=4, d=5. Sâ‚‚â‚€=10[8+95]=1030

Q8: Find sum of first 51 terms where 2nd=14 and 3rd=18. d=4, a=10. S₅₁=5610

Q9: Sum of first 7 terms=49, sum of first 17 terms=289. Find sum of first n terms. â†' a=3, d=2. Sâ‚™=n(n+2)

Q11: Sum of first n terms of AP is 4nâˆ'n². Find a, d, and Sâ‚™ formula. â†' Compare: Sâ‚™=n/2[2a+(nâˆ'1)d] with n(4âˆ'n). a=3, d=âˆ'2.

Q18: Spiral made of semicircles with radii 0.5, 1.0, 1.5, 2.0 cm... Total length of 13 consecutive semicircles. â†' Ï€/2(0.5+1.0+1.5+...+6.5) = Ï€/2 × S₁₃. AP: 0.5,1,1.5,... d=0.5. S₁₃=13/2[1+12×0.5]=13×3.5=45.5×π/2 cm

Q20: 200 logs stacked: 20 bottom, 19, 18... How many rows? Sâ‚™=n/2[40âˆ'(nâˆ'1)]=200 â†' 20nâˆ'n(nâˆ'1)/2=200 â†' n²âˆ'41n+400=0 â†' n=16 or 25

Which term of AP 121, 117, 113... is its first negative term? a=121, d=âˆ'4. aâ‚™<0 â†' 121âˆ'4(nâˆ'1)<0 â†' n>31.25. n=32. 32nd term = âˆ'3

A ladder has rungs 25 cm apart. Top rung=45 cm, bottom=25 cm. Wood required for 2.5 m ladder? â†' Number of rungs=250/25+1=11. Sum formula for AP: a=25, l=45, n=11. S₁₁=11/2(25+45)=385 cm