0NIAN'S LEARNING HUB - CBSE MATHS X
Nidhi Sharma
Arithmetic Progressions
Examination Assignment + Stepwise Answer Key
Instructions
Attempt without seeing the answer key.
Show formula, substitution and simplification. Marks are often lost when steps are skipped.
Questions marked TRAP are designed around common sign, n−1, wording and term-vs-sum mistakes.
Total: 25 questions. Suggested time: 60–75 minutes.
Part A — Core Formula Practice
1. Find the common difference of the AP: 18, 13, 8, 3, …
2. Write the first four terms of an AP with a = −3 and d = 5.
3. Find the 20th term of the AP: 7, 11, 15, …
4. Which term of the AP 5, 9, 13, … is 81?
5. Find the number of terms in 3, 8, 13, …, 98.
6. Find the 8th term from the end of the AP 4, 9, 14, …, 124.
7. Find the sum of the first 30 terms of the AP 2, 5, 8, …
8. Find the sum of the AP 7, 12, 17, …, 102.
9. If the 7th term of an AP is 20 and the 13th term is 38, find a and d.
10. The 4th term of an AP is 12 and the 9th term is 27. Find the 20th term.
Part B — Formula Selection & Examination Questions
11. The sum of the first n terms is Sₙ = 3n² + 2n. Find the nth term and the common difference.
12. Find the sum of the first 25 positive multiples of 6.
13. Find the sum of all two-digit natural numbers divisible by 7.
14. Three numbers are in AP. Their sum is 24 and their product is 440. Find the numbers.
15. Four numbers are in AP. Their sum is 32 and the product of the first and fourth terms is 55. Find the numbers.
16. The 5th term of an AP is 3 times its 2nd term. If the 8th term is 37, find the AP.
17. How many terms of the AP 24, 21, 18, … must be taken so that their sum is 78?
18. A theatre has 20 seats in the first row, 24 in the second, 28 in the third and so on. How many seats are there in the 15th row? How many seats are there in the first 15 rows?
Part C — TRAP Questions
19. TRAP Is 7, 4, 1, −2, … an AP? If yes, state d.
20. TRAP Find the 15th term of the AP 10, 7, 4, 1, …
21. TRAP Which term of 2, 7, 12, 17, … is 54?
22. TRAP Find the 6th term from the end of 5, 8, 11, …, 95.
23. TRAP The sum of the first n natural numbers is 210. Find n.
24. TRAP The 4th and 8th terms of an AP add to 32, and the 6th and 10th terms add to 52. Find the first term and common difference.
25. TRAP For an AP, Sₙ = 2n² + 5n. Find a₁, a₂, d and a₁₀.
STEPWISE ANSWER KEY
Use these solutions to check the method, not only the final answer.
1.
Given: 18, 13, 8, 3, …
d = 13 − 18 = −5
Check: 8 − 13 = −5
Answer: d = −5.
2.
a = −3, d = 5
a₁ = −3
a₂ = a+d = −3+5 = 2
a₃ = a+2d = −3+10 = 7
a₄ = a+3d = −3+15 = 12
Answer: −3, 2, 7, 12.
3.
a = 7, d = 4, n = 20
aₙ = a+(n−1)d
a₂₀ = 7+19×4 = 7+76 = 83
Answer: 83.
4.
a = 5, d = 4, aₙ = 81
81 = 5+(n−1)4
76 = 4(n−1)
19 = n−1
n = 20
Answer: 81 is the 20th term.
5.
a = 3, d = 5, last term = 98
98 = 3+(n−1)5
95 = 5(n−1)
19 = n−1
n = 20
Answer: 20 terms.
6.
AP: 4, 9, 14, …, 124; d = 5; l = 124
8th term from end = l−(8−1)d
= 124−7×5 = 124−35 = 89
Answer: 89.
7.
a = 2, d = 3, n = 30
Sₙ = n/2[2a+(n−1)d]
S₃₀ = 30/2[4+29×3]
= 15(4+87) = 15×91 = 1365
Answer: 1365.
8.
a = 7, d = 5, l = 102
First find n: 102 = 7+(n−1)5
95 = 5(n−1) ⇒ n−1 = 19 ⇒ n = 20
Sₙ = n/2(a+l)
S₂₀ = 20/2(7+102) = 10×109 = 1090
Answer: 1090.
9.
a₇ = a+6d = 20 …(1)
a₁₃ = a+12d = 38 …(2)
Subtract (1) from (2): 6d = 18 ⇒ d = 3
a+18 = 20 ⇒ a = 2
Answer: a = 2, d = 3.
10.
a₄ = a+3d = 12 …(1)
a₉ = a+8d = 27 …(2)
Subtract: 5d = 15 ⇒ d = 3
a+9 = 12 ⇒ a = 3
a₂₀ = 3+19×3 = 60
Answer: 60.
11.
Sₙ = 3n²+2n
aₙ = Sₙ−Sₙ₋₁
Sₙ₋₁ = 3(n−1)²+2(n−1) = 3n²−4n+1
aₙ = (3n²+2n)−(3n²−4n+1) = 6n−1
So a₁ = 5 and a₂ = 11
d = 11−5 = 6
Answer: aₙ = 6n−1, d = 6.
12.
Positive multiples of 6: 6, 12, 18, …
a = 6, d = 6, n = 25
S₂₅ = 25/2[12+24×6]
= 25/2(156) = 25×78 = 1950
Answer: 1950.
13.
Two-digit multiples of 7: 14, 21, 28, …, 98
a = 14, d = 7, l = 98
98 = 14+(n−1)7 ⇒ 84 = 7(n−1) ⇒ n = 13
S₁₃ = 13/2(14+98) = 13/2×112 = 728
Answer: 728.
14.
Let the three AP terms be a−d, a, a+d.
Sum: 3a = 24 ⇒ a = 8.
Product: (8−d)·8·(8+d) = 440
8(64−d²) = 440 ⇒ 64−d² = 55
d² = 9 ⇒ d = 3 (or −3, same set in reverse order)
Answer: 5, 8, 11.
15.
Let four AP terms be a−3d, a−d, a+d, a+3d (common difference = 2d).
Sum = 4a = 32 ⇒ a = 8.
First × fourth = (8−3d)(8+3d) = 55
64−9d² = 55 ⇒ 9d² = 9 ⇒ d = 1
Terms: 5, 7, 9, 11.
Answer: 5, 7, 9, 11.
16.
a₅ = a+4d; a₂ = a+d
Given a+4d = 3(a+d)
a+4d = 3a+3d ⇒ d = 2a …(1)
Also a₈ = a+7d = 45 …(2)
Using d=2a: a+14a = 45 ⇒ 15a = 45
a = 3, d = 6
Answer: AP is 3, 9, 15, 21, … (common difference 6).
17.
a = 24, d = −3, Sₙ = 78
78 = n/2[48+(n−1)(−3)]
156 = n[48−3n+3] = n(51−3n)
156 = 51n−3n²
3n²−51n+156 = 0
Divide by 3: n²−17n+52 = 0
(n−4)(n−13)=0
n = 4 or 13
Answer: 4 terms or 13 terms (both give sum 78).
18.
Seats form AP: 20, 24, 28, …; a=20, d=4
15th row: a₁₅ = 20+14×4 = 76 seats
Total first 15 rows: S₁₅ = 15/2[40+14×4]
= 15/2(96) = 720
Answer: 76 seats in 15th row; 720 seats in first 15 rows.
19. TRAP
Check consecutive differences:
4−7 = −3, 1−4 = −3, −2−1 = −3
All differences are equal.
Answer: Yes, it is an AP; d = −3.
20. TRAP
a = 10, d = −3, n = 15
a₁₅ = a+(15−1)d
= 10+14(−3) = 10−42 = −32
Trap: keep brackets around negative d.
Answer: −32.
21. TRAP
a = 2, d = 5; set aₙ = 54
54 = 2+(n−1)5
52 = 5(n−1)
n−1 = 52/5 = 10.4
n = 11.4, not a positive integer term number.
Answer: 54 is NOT a term of this AP.
22. TRAP
l = 95, d = 3
6th term from end = l−(6−1)d
= 95−5×3 = 80
Trap: 6th from end means 5 jumps backward, not 6.
Answer: 80.
23. TRAP
1+2+…+n = n(n+1)/2 = 210
n(n+1)=420
n²+n−420=0
(n−20)(n+21)=0
Positive solution n=20
Answer: 20.
24. TRAP
Important wording: '4th and 8th terms add to 32' means a₄+a₈=32, not a₄=32.
a₄+a₈ = (a+3d)+(a+7d)=2a+10d=32 …(1)
a₆+a₁₀ = (a+5d)+(a+9d)=2a+14d=52 …(2)
Subtract: 4d=20 ⇒ d=5
From (1): 2a+50=32 ⇒ 2a=−18 ⇒ a=−9
Answer: a = −9, d = 5.
25. TRAP
Sₙ = 2n²+5n
a₁ = S₁ = 2+5 = 7
a₂ = S₂−S₁ = (8+10)−7 = 11
d = a₂−a₁ = 4
Or aₙ = Sₙ−Sₙ₋₁ = 4n+3
a₁₀ = 4(10)+3 = 43
Answer: a₁=7, a₂=11, d=4, a₁₀=43.
Final Self-Check Before Submitting
Did I write d = next − previous?
Did I use (n−1), not n?
Did I distinguish aₙ (term) from Sₙ (sum)?
Did I
keep brackets around a negative common difference?
If I solved for n, is n a positive integer?
For 'from the end', did I count jumps correctly?
If two terms 'add to' a number, did I write aᵣ + aₛ = that number?
