Arithmetic Progressions – Question Bank
A. Multiple Choice Questions (MCQs)
1. Which of the following is an arithmetic progression?
A. 2, 4, 8, 16, …
B. 3, 7, 11, 15, …
C. 1, 2, 4, 7, …
D. 5, 10, 20, 40, …
Answer: B
2. The common difference of the AP 17, 12, 7, 2, … is:
A. 5
B. −5
C. 7
D. −7
Answer: B
3. If the first term of an AP is 6 and its common difference is 4, the third term is:
A. 10
B. 12
C. 14
D. 16
Answer: C
4. The nth term of an AP with first term a and common difference d is:
A. a + nd
B. a + (n + 1)d
C. a + (n − 1)d
D. n(a + d)
Answer: C
5. The 15th term of the AP 4, 9, 14, 19, … is:
A. 69
B. 74
C. 79
D. 84
Answer: B
6. Which sequence is NOT an AP?
A. 5, 8, 11, 14, …
B. 20, 17, 14, 11, …
C. −3, 0, 3, 6, …
D. 2, 4, 8, 16, …
Answer: D
7. If an AP has first term 10 and common difference −2, its fifth term is:
A. 0
B. 2
C. 4
D. 8
Answer: C
8. If the nth term of an AP is 25 and the first term is 5 with common difference 4, then n is:
A. 5
B. 6
C. 7
D. 8
Answer: B
9. The sum of the first n terms of an AP is:
A. n[2a + (n − 1)d]
B. n/2[2a + (n − 1)d]
C. 2n[2a + (n − 1)d]
D. n/2[a + (n − 1)d]
Answer: B
10. If the first term is 7, the last term is 37 and there are 6 terms, their sum is:
A. 120
B. 126
C. 132
D. 144
Answer: C
11. If three numbers a, b and c are in AP, then:
A. a + b = c
B. a + c = b
C. 2b = a + c
D. 2a = b + c
Answer: C
12. The common difference of the AP −4, −1, 2, 5, … is:
A. −3
B. 3
C. 2
D. −2
Answer: B
13. Which of the following can be the common difference of an AP?
A. Only positive numbers
B. Only negative numbers
C. Only non-zero numbers
D. Positive, negative or zero
Answer: D
14. The 20th term of the AP 3, 6, 9, 12, … is:
A. 57
B. 60
C. 63
D. 66
Answer: B
15. If Sₙ denotes the sum of the first n terms, then the nth term is:
A. Sₙ + Sₙ₋₁
B. Sₙ − Sₙ₋₁
C. Sₙ × Sₙ₋₁
D. Sₙ/Sₙ₋₁
Answer: B
16. The sum 1 + 2 + 3 + … + 50 is:
A. 1250
B. 1275
C. 1300
D. 1325
Answer: B
17. If a = 8, d = 3 and n = 10, then aₙ is:
A. 32
B. 35
C. 38
D. 41
Answer: C
18. If an AP has a = 12, d = 5 and n = 20, its last term is:
A. 102
B. 107
C. 112
D. 117
Answer: C
19. The sum of the first 10 terms of 5, 10, 15, … is:
A. 250
B. 275
C. 300
D. 325
Answer: B
20. If the first term of an AP is 15 and the common difference is 0, the AP is:
A. 15, 16, 17, …
B. 15, 14, 13, …
C. 0, 15, 30, …
D. 15, 15, 15, …
Answer: D
B. Fill in the Blanks
1.
An arithmetic progression is a sequence in which the difference between consecutive terms is __________.
Answer: constant
2.
The first term of an AP is generally represented by __________.
Answer: a
3.
The common difference of an AP is represented by __________.
Answer: d
4.
For an AP, d = __________ − previous term.
Answer: next term
5.
The general form of an AP is a, a + d, __________, __________, …
Answer: a + 2d, a + 3d
6.
The nth term of an AP is given by aₙ = __________.
Answer: a + (n − 1)d
7.
The sum of the first n terms of an AP is denoted by __________.
Answer: Sₙ
8.
Sₙ = n/2 [__________].
Answer: 2a + (n − 1)d
9.
If l is the last term of a finite AP, then Sₙ = __________.
Answer: n/2(a + l)
10.
If a, b and c are in AP, then __________ = a + c.
Answer: 2b
11.
The common difference of 8, 8, 8, 8, … is __________.
Answer: 0
12.
The sequence 12, 9, 6, 3, … has common difference __________.
Answer: −3
13.
The first term of the AP 5, 9, 13, 17, … is __________.
Answer: 5
14.
The common difference of 5, 9, 13, 17, … is __________.
Answer: 4
15.
The nth term of an AP can also be written as the difference between __________ and __________.
Answer: Sₙ and Sₙ₋₁
C. True or False
1.
The common difference of an AP must always be positive.
Answer: False
2.
A sequence with common difference zero is an AP.
Answer: True
3.
The sequence 2, 5, 8, 11, … is an AP.
Answer: True
4.
The sequence 2, 4, 8, 16, … is an AP.
Answer: False
5.
In an AP, consecutive terms have the same difference.
Answer: True
6.
The nth term of an AP is a + nd.
Answer: False
7.
The formula for the nth term is a + (n − 1)d.
Answer: True
8.
A decreasing sequence can never be an AP.
Answer: False
9.
If three numbers are in AP, twice the middle number equals the sum of the other two.
Answer: True
10.
Sₙ represents the nth term of an AP.
Answer: False
11.
A finite AP has a last term.
Answer: True
12.
An AP can have a negative common difference.
Answer: True
D. Very Short Answer Questions
1.
What is an arithmetic progression?
Answer: A sequence in which the difference between consecutive terms is constant.
2.
What is the common difference of 4, 9, 14, 19, …?
Answer: 5
3.
Write the general form of an AP.
Answer: a, a + d, a + 2d, a + 3d, …
4.
Write the formula for the nth term of an AP.
Answer: aₙ = a + (n − 1)d
5.
What does a represent in an AP?
Answer: The first term.
6.
What does d represent?
Answer: The common difference.
7.
What does Sₙ represent?
Answer: The sum of the first n terms.
8.
What is the common difference of 7, 7, 7, 7, …?
Answer: 0
9.
What is the 1st term of an AP?
Answer: a
10.
What is the formula for the sum of the first n terms when the last term is l?
Answer: Sₙ = n/2(a + l)
E. Short Answer Questions
1.
Determine whether 4, 9, 14, 19, 24, … is an AP. Give a reason.
Answer: Yes. The consecutive differences are all 5.
2.
Determine whether 2, 5, 10, 17, … is an AP.
Answer: No. The consecutive differences are 3, 5 and 7, which are not equal.
3.
Find the first four terms of an AP whose first term is 7 and common difference is −3.
Answer: 7, 4, 1, −2
4.
Find the first term and common difference of:
18, 13, 8, 3, …
Answer:
a = 18
d = −5
5.
Find the 12th term of:
5, 8, 11, 14, …
Answer: 38
6.
Find the 20th term of:
12, 9, 6, 3, …
Answer: −45
7.
Which term of the AP 7, 11, 15, 19, … is 63?
Answer: 15th term
8.
Check whether 100 is a term of the AP 4, 10, 16, 22, …
Answer: No. The required value of n is not an integer.
9.
Find the sum of the first 15 terms of:
3, 7, 11, 15, …
Answer: 495
10.
Find the sum of the first 20 positive integers.
Answer: 210
11.
Find the arithmetic mean of 12 and 28.
Answer: 20
12.
Find the number that must be inserted between 9 and 21 so that the three numbers form an AP.
Answer: 15
F. Assertion and Reason Questions
For each question, choose:
A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, but Reason is false.
D. Assertion is false, but Reason is true.
1.
Assertion: 5, 10, 15, 20, … is an AP.
Reason: The difference between consecutive terms is always 5.
Answer: A
2.
Assertion: 10, 7, 4, 1, … is an AP.
Reason: Its common difference is −3.
Answer: A
3.
Assertion: 2, 4, 8, 16, … is an AP.
Reason: Each term is obtained by multiplying the previous term by 2.
Answer: D
4.
Assertion: A constant sequence can be an AP.
Reason: Its common difference is zero.
Answer: A
5.
Assertion: The nth term of an AP is a + (n − 1)d.
Reason: The first term contains zero multiples of d added to a.
Answer: A
6.
Assertion: If three numbers are in AP, their middle term is the arithmetic mean of the other two.
Reason: For three terms in AP, the difference between consecutive terms is equal.
Answer: A
G. Find the Missing Terms
1.
Complete the AP:
4, ___, 12, 16, …
Answer: 8
2.
Complete:
20, 15, ___, 5, …
Answer: 10
3.
Complete:
−6, −2, ___, 6, …
Answer: 2
4.
Complete:
3, ___, ___, 15
Answer: 7, 11
5.
Complete:
25, 20, ___, ___, 5
Answer: 15, 10
6.
Complete:
___, 12, 17, 22, …
Answer: 7
H. Case-Based Questions
Case Study 1: Monthly Savings
A student saves Rs. 100 in the first month and increases the monthly saving by Rs. 50 every month.
The savings are:
100, 150, 200, 250, …
Answer the following:
(a)
Is this sequence an AP?
Answer: Yes.
(b)
What is the first term?
Answer: Rs. 100
(c)
What is the common difference?
Answer: Rs. 50
(d)
How much will be saved in the 10th month?
Answer: Rs. 550
(e)
What is the total saving during the first 10 months?
Answer: Rs. 3,250
Case Study 2: Production
A factory produces 500 units in the first year and increases its production by 25 units every year.
(a)
Write the first three terms.
Answer: 500, 525, 550
(b)
Find the common difference.
Answer: 25
(c)
Find the production in the 10th year.
Answer: 725 units
(d)
Find the total production during the first 10 years.
Answer: 6,125 units
Case Study 3: Rows of Plants
A garden has 30 plants in the first row, 27 in the second, 24 in the third, and so on.
(a)
What is the common difference?
Answer: −3
(b)
Write the general term.
Answer: aₙ = 30 + (n − 1)(−3)
(c)
How many plants are in the 8th row?
Answer: 9
(d)
Is the number of plants increasing or decreasing?
Answer: Decreasing.
I. Competency-Based Questions
1.
A teacher writes:
11, 16, 21, 26, …
A student says that the 25th term is 131.
Is the student’s answer correct? Show the calculation.
Answer: Yes.
a = 11, d = 5
a₂₅ = 11 + 24(5)
= 131
2.
A sequence has first term 18 and common difference −4.
Without writing all its terms, determine whether 2 occurs in the sequence.
Answer: Yes.
2 = 18 + (n − 1)(−4)
Solving gives n = 5.
Therefore, 2 is the 5th term.
3.
Two students use different formulas to find the sum of an AP.
Student A uses:
Sₙ = n/2 [2a + (n − 1)d]
Student B uses:
Sₙ = n/2(a + l)
Can both formulas be correct?
Answer: Yes. The first uses the common difference, while the second uses the last term.
4.
Three consecutive terms of an AP are:
x − 4, x, x + 4
Find x if their sum is 36.
Answer:
(x − 4) + x + (x + 4) = 36
3x = 36
x = 12
The terms are:
8, 12, 16
J. Higher-Order Thinking Questions
1.
The 5th term of an AP is 18 and the 9th term is 30. Find the first term and common difference.
Answer: a = 6, d = 3
2.
The 7th term of an AP is 34 and the 13th term is 58. Find the 20th term.
Answer: 86
3.
The 4th and 8th terms of an AP have a sum of 30. The 6th and 10th terms have a sum of 46. Find the first term and common difference.
Answer: a = 2, d = 2
4.
Find the number of terms in:
8, 13, 18, …, 103
Answer: 20
5.
How many multiples of 5 lie between 10 and 100?
Answer: 17
6.
Find the sum of all multiples of 4 from 4 to 100.
Answer: 1,300
7.
The first term of an AP is 5, its last term is 45, and its sum is 400. Find the number of terms.
Answer: 16
8.
An AP has 20 terms. Its first term is 7 and its last term is 64. Find its common difference.
Answer: 3
K. Mixed Practice Questions
1.
Find the 18th term of:
7, 12, 17, 22, …
2.
Find the 25th term of:
30, 27, 24, 21, …
3.
Which term of:
4, 9, 14, 19, …
is 99?
4.
Check whether 150 is a term of:
6, 12, 18, 24, …
5.
Find the sum of the first 25 terms of:
2, 6, 10, 14, …
6.
Find the sum of:
15 + 12 + 9 + … + (−12)
7.
Find three consecutive terms of an AP whose sum is 42 and common difference is 5.
8.
Find the first term of an AP whose 8th term is 31 and common difference is 4.
9.
If the 3rd term of an AP is 12 and the 10th term is 40, find the common difference.
10.
If the sum of the first n terms of an AP is known, explain how the nth term can be obtained.
L. Answer Key – Quick Check
MCQs
1-B
2-B
3-C
4-C
5-B
6-D
7-C
8-B
9-B
10-C
11-C
12-B
13-D
14-B
15-B
16-B
17-C
18-C
19-B
20-D
Fill in the Blanks
- constant
- a
- d
- next term
- a + 2d, a + 3d
- a + (n − 1)d
- Sₙ
- 2a + (n − 1)d
- n/2(a + l)
- 2b
- 0
- −3
- 5
- 4
- Sₙ, Sₙ₋₁
True/False
1-F
2-T
3-T
4-F
5-T
6-F
7-T
8-F
9-T
10-F
11-T
12-T
Most Important Questions for Exam Preparation
If students have limited time, these question types should be prioritised:
- Identify whether a sequence is an AP.
- Find the first term and common difference.
- Find the nth term.
- Find which term contains a given number.
- Find an AP when two terms are given.
- Find the number of terms when the first and last terms are known.
- Find the sum of the first n terms.
- Solve word problems involving a fixed increase/decrease.
- Use the relationship aₙ = Sₙ − Sₙ₋₁.
- Apply the arithmetic-mean property to three terms in AP.
- Solve case-based AP questions.
- Solve questions involving multiples, consecutive integers, rows, savings, production and similar real-life patterns.