Practice Questions on Arithmetic Progression (Basic)

An arithmetic progression (AP), also known as an arithmetic sequence, is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is called the "common difference."

Important Formulas on Arithmetic Progression

Various formulas on arithmetic progression are:

where,

• a_n is the nth term
• a₁ is the first term
• d is the common difference
• S_n is the sum of the first n terms
• S_n-1 is the sum of the first n-1 terms
• n is the number of terms

Practice Questions on Arithmetic Progression with Solution

Question 1: Find the common difference from the given AP: 1, 6, 11, 16, 21.

Solution:

To find common difference we need to subtract consecutive terms

d = a_n - a_n-1

d = 6 - 1 = 5
So, common difference of the given AP is 5.

Question 2: If the 4th term of an AP is 8 with a common difference of 2. Find out the arithmetic progression up to 8 terms.

Solution:

Given that, the fourth term, a4 is 8 and the common difference is 2,

So the fourth term can be written as,
a + (4 - 1) × 2 = 8  [a = first term]
= a + 3 × 2 = 8
= a = 8 - 3 × 2
= a = 8 - 6
= a = 2

So the first term a₁ is 2,
Now, a₂ = a₁ + 2 = 2 + 2 = 4 similarly the series upto 8 term is = 2, 4, 6, 8, 10, 12, 14, 16.

Question 3: Find the 40th term in the 2, 5, 8, 11, 14,... sequence

Solution:

a = 2
d = 5 - 2 = 3
n = 40

a_n = a + ( n - 1 )d
a₄₀ = 2 + ( 40 - 1 )3
= 2 + 117
a₄₀ = 119

Question 4: Find the 9th term in the 1/3, 2/3, 1, 4/3... sequence

Solution:

a = 1/3
d = 2/3 - 1/3 = 1/3 
n = 9

a_n = a + ( n -1 )d

a₉ = 1/3 + ( 9 - 1 )(1/3)
= 1/3 + 8/3
= 9/3 = 3

Question 5: Find the 15th term of the sequence 9, 7, 5, 3, 1,...

Solution:

a = 9
d = 7 - 9 = 5 - 7 = -2
n = 15

a₁₅ = a + ( n - 1 ) × d
= 9 + ( 15 - 1 ) × (-2)
= 9 + ( 14 ) × ( -2 )
= 9 - 28
a₁₅ = -21

Hence, 15th term in the sequence 9,7,5,3,1,... is -21

Question 6: Find the general term of the series 4,7, 10,13......

Solution:

In the given AP, we have
a₁ = 4, d = 3

So, general term of the given AP is
a_n = a₁ + (n - 1)d
a_n = 4 + (n - 1)3
a_n = 4 + 3n - 3
a_n = 3n + 1

General term = 3n + 1.

Question 7: Find the 50th term of the sequence 293, 290, 287, 284, 281,...

Solution:

a = 293
d = 290 - 293 = 287 - 290 = -3
n = 50

a₅₀  = a + (n - 1) × d
= 293 + ( 50 - 1 ) × (-3)
= 293 + ( 49 ) × ( -3 )
= 293 - 147
a₅₀ = 146

Hence, 50th term in the sequence 293, 290, 287, 284, 281,... is 146

Question 8: Find the 100th term of the sequence 100, 50, 0, -50, -100,...

Solution:

a = 100
d = 50 - 100 = 0 - 50 = -50
n = 100

a₁₀₀  = a + ( n - 1 ) × d
= 100 + ( 100 - 1 ) × (-50)
= 100 + ( 99 ) × ( -50 )
= 100 - 4950
a₁₀₀ = -4850

Hence, 100th term in the sequence 100, 50, 0, -50, -100,... is -4850

Question 9: What is the sum of the first 20 odd natural numbers?

Solution.

First odd natural number is 1
Common Difference is 2

So, 20^th term is

a₂₀ = a₁ + (n-1)d

a₂₀ = 1 + (20-1)2
a₂₀ = 1 + 38
a₂₀ = 39
So,

Sum = n/2(a₁ + a₂₀)

Sum = 20/2(1 + 39)
Sum = 10 × 40
Sum = 400

So, sum of first 20 odd numbers = 400

Question 10: Which term of the arithmetic progression 30, 27, 24,.... is 0?

Solution:

In the given AP,

a₁ = 30
d = -3
a_n = 0

Now, to find n

a_n = a₁ + (n-1)d

0 = 30 + (n - 1)(-3)
0 = 30 - 3n + 3
3n = 33
n = 11

So, the 11th term is 0

Question 11: Find the first negative term of the arithmetic progression 36, 30, 24,.....?

Solution:

In the given AP, we have

a₁ = 36
d = -6

To find the first negative term,

36 + (n−1)(−6) < 0
36 + 6 - 6n < 0
42 - 6n < 0
-6n < - 42
n > 7

So, the 8th term is the first negative term of the given AP.

Question 12: If the common difference of an A.P. is 4, then a₂₀ - a₁₅ is?

Solution:

So, the given common difference = 4.

Now, we want to calculate a20 - a15,
So,
a₂₀ = a + 19d
a₁₅ = a + 14d

Then,
a₂₀ - a₁₅ = a+ 19d - a - 14d
a₂₀ - a₁₅ = 5d
a₂₀ - a₁₅ = 5 × 4 = 20

So the value of a₂₀ - a₁₅ is 20.

Question 13: Which term of the arithmetic progression 3, 8, 13, ..... is 55 more than its 20th term?

Solution:

In the given AP, we have following values

a = 3
d = 5

Let the unknown term is n term,

Now,
a_n = a₂₀ + 55
a₁ + (n - 1)d = a1 + 19d + 55
3 + (n-1)d = 3 + 19×5 + 55
5n - 5 = 95 + 55
5n = 155
n = 31

So, the 31st term is 55 more than the 20th term.

Question 14: Determine the sum of the first 18 terms, of the arithmetic progression 12b, 8b, 4b,......

Solution:

In the given AP we have,

a₁ = 12b

d = -4b

So, the sum of first 18 terms is

Sum = n/2[2a₁ + (n - 1)d]
Sum = 9[2 × 12b + (18 - 1)(-4b)]
Sum = 9[24b - 68b]
Sum = 9 × (-44b)
Sum = -396b

So sum of first 18 terms is -396b.

Question 15: If the 3rd term of an arithmetic sequence is 12 and the 7th term is 24, what is the first term?

Solution:

The nth term : an = a+ (n-1) d

a₃ = 12 = a + 2d - (1)
a₇ = 24 = a + 6d - (2)

Subtract (1) from (2), (2) - (1) :

a₇ - a3 = 24 - 12 = 12 = a + 6d - a - 2d = 6d - 2d = 4d
12 = 4d
d = 12 / 4 = 3

Put value of d in (1) :
12 = a + 2(3)
a = 12 - 6 = 6

Hence the 1st term of arithmetic sequence is 6.

Practice Questions on Arithmetic Progression (Unsolved)

Question 1: In an AP, if d = 8, n = 10, and a_n = 80, then Find the value of a.

Question 2: Find the 19th term of the AP: 5, 11, 17, 23, ...?

Question 3: What is the sum of first 30 even natural numbers?

Question 4: Which term of the arithmetic progression 13, 18, 23, ..... is 80 more than its 20th term?

Question 5: If the sum of first p terms of an A.P., is ap² + bp, find its common difference.

Question 6: The first term of an A.P. is p and its common difference is q. Find its 10th term

Question 7: If seven times the 7th term of an A.P. is equal to eleven times the 11th term, then what will be its 18th term?

Question 8: How many terms of the A.P. 18, 16, 14, … … be taken so that their sum is zero?

Question 9: 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three times its 11th term.

Question 10: Find the First Term Given a Condition:

• The 8th term of an AP is zero, and the 12th term is 20. Find the first term and the common difference.
• If the 5th term of an AP is 15 and the sum of the first 10 terms is 120, find the first term.

Answer key

1. a = 8
2. 19th term of the AP: 113
3. Sum of first 30 even natural numbers: 930
4. 36th term is 80 more than the 20th term.
5. Common difference d = 2a
6. 10th term of the AP: p + 9q
7. 18th term is 34d
8. 19 terms should be taken.
9. Proof: 25th term = 3 × 11th term
10. Middle term: 9
    Middle term: 27

Also Check:

• Practice Problems on Finite and Infinite Sets
• Practice Questions on Addition and Subtraction of Rational Numbers
• Mean Median Mode Practice Questions
• Practice Questions on Data Handling
• Practice Problems on Complement of a Set
• Union and Intersection of Sets Practice Questions
• Area of Rectangle Questions
• Practice Problem on Surface Area and Volume Formulas