CBSE Class 10 Mathematics (2026–27)
Chapter 5: Arithmetic Progressions (AP)
20 Important Questions and Answers
Q1. What is an Arithmetic Progression? Give two examples.
Answer:
An Arithmetic Progression (AP) is a sequence of numbers in which the difference between any two consecutive terms remains constant. This constant difference is called the common difference (d). If the first term is represented by a, then an AP can be written as: a, a + d, a + 2d, a + 3d, … . Arithmetic Progressions are commonly used in mathematics to study patterns and sequences. For example, 2, 5, 8, 11, 14, … is an AP with common difference 3. Another example is 20, 16, 12, 8, 4, … where the common difference is –4. APs help in solving problems related to series and patterns.
Q2. Define the first term and common difference of an AP.
Answer:
In an Arithmetic Progression, the first term (a) is the number with which the sequence begins. The common difference (d) is the fixed value added to or subtracted from each term to obtain the next term. The common difference can be found by subtracting any term from the next term. For example, in the AP 7, 12, 17, 22, 27, …, the first term is 7 and the common difference is 12 – 7 = 5. Every term of the sequence is obtained by adding 5 to the previous term. The concepts of first term and common difference are essential for finding any term of an AP.
Q3. Find the common difference of the AP: 15, 11, 7, 3, –1.
Answer:
To find the common difference of an Arithmetic Progression, subtract a term from the term immediately following it. In the given AP:
15, 11, 7, 3, –1
d = 11 – 15 = –4
Checking further:
7 – 11 = –4
3 – 7 = –4
–1 – 3 = –4
Since the difference between consecutive terms remains the same, the sequence is an Arithmetic Progression. Therefore, the common difference (d) = –4. A negative common difference indicates that the terms are decreasing. Such APs are called decreasing arithmetic progressions. The common difference helps determine future terms and identify the pattern followed in the sequence.
Q4. Write the first four terms of an AP whose first term is 8 and common difference is 6.
Answer:
The first term of the AP is given as a = 8 and the common difference is d = 6. Each successive term is obtained by adding 6 to the previous term.
First term = 8
Second term = 8 + 6 = 14
Third term = 14 + 6 = 20
Fourth term = 20 + 6 = 26
Thus, the first four terms of the AP are:
8, 14, 20, 26
Arithmetic Progressions are formed by repeatedly adding the common difference to the previous term. This method can be used to generate as many terms as required. Such sequences are widely used in mathematics and real-life situations involving regular increments.
Q5. What is the nth term of an Arithmetic Progression?
Answer:
The nth term of an Arithmetic Progression is the term located at the nth position in the sequence. It is represented by an and can be calculated using the formula:
an = a + (n – 1)d
where:
- a = first term
- d = common difference
- n = term number
This formula helps find any term directly without writing all previous terms. For example, if a = 3 and d = 4, then the 10th term is:
a10 = 3 + (10 – 1) × 4
= 3 + 36
= 39
Thus, the nth term formula is a powerful tool for solving problems involving arithmetic progressions quickly and efficiently.
Q6. Find the 15th term of the AP: 4, 9, 14, 19, …
Answer:
Given:
First term, a = 4
Common difference, d = 9 – 4 = 5
Term number, n = 15
Using the nth term formula:
an = a + (n – 1)d
a15 = 4 + (15 – 1) × 5
= 4 + 14 × 5
= 4 + 70
= 74
Therefore, the 15th term of the AP is 74. This method allows us to find any term directly without listing all the terms. Arithmetic Progressions are useful in solving various mathematical and practical problems involving regularly increasing or decreasing values.
Q7. Determine whether 45 is a term of the AP: 5, 9, 13, 17, …
Answer:
Given:
a = 5
d = 4
We need to check whether 45 is a term of the AP.
Using the nth term formula:
45 = 5 + (n – 1) × 4
40 = 4(n – 1)
10 = n – 1
n = 11
Since n is a positive integer, 45 is the 11th term of the AP. Therefore, 45 belongs to the given Arithmetic Progression. This method is useful for verifying whether a particular number occurs in an AP without writing all the terms.
Q8. Find the 20th term of the AP: 7, 10, 13, 16, …
Answer:
Given:
a = 7
d = 10 – 7 = 3
n = 20
Using the formula:
an = a + (n – 1)d
a20 = 7 + (20 – 1) × 3
= 7 + 19 × 3
= 7 + 57
= 64
Therefore, the 20th term of the AP is 64. Arithmetic Progression formulas help find distant terms quickly. Instead of writing twenty terms manually, the nth term formula provides an efficient solution. Such applications are common in mathematics and competitive examinations.
Q9. Find the first term if the 8th term of an AP is 29 and the common difference is 3.
Answer:
Given:
a8 = 29
d = 3
Using the formula:
an = a + (n – 1)d
29 = a + (8 – 1) × 3
29 = a + 21
a = 29 – 21
a = 8
Therefore, the first term of the AP is 8. Knowing any term and the common difference allows us to determine the first term. This concept is useful when some information about an AP is provided and the starting value needs to be found.
Q10. Find the common difference if the first term is 12 and the 10th term is 57.
Answer:
Given:
a = 12
a10 = 57
Using the nth term formula:
57 = 12 + (10 – 1)d
57 = 12 + 9d
45 = 9d
d = 5
Therefore, the common difference is 5. The common difference represents the fixed change between consecutive terms. Once it is known, the entire arithmetic progression can be constructed. This concept is fundamental in solving AP-related problems.
Q11. Explain how an AP can be represented algebraically.
Answer:
An Arithmetic Progression can be represented algebraically using the first term and common difference. If the first term is a and the common difference is d, then the sequence is:
a, a + d, a + 2d, a + 3d, …
The nth term is represented by:
an = a + (n – 1)d
This algebraic representation helps in finding any term directly without writing the entire sequence. It simplifies calculations and allows mathematicians to solve large problems efficiently. APs are used in various fields such as finance, engineering, and statistics where regular patterns occur.
Q12. Find the 25th term of the AP: 2, 7, 12, 17, …
Answer:
Given:
a = 2
d = 7 – 2 = 5
n = 25
Using:
an = a + (n – 1)d
a25 = 2 + (25 – 1) × 5
= 2 + 24 × 5
= 2 + 120
= 122
Therefore, the 25th term is 122. The nth term formula helps determine any term quickly. Arithmetic Progressions are useful in understanding number patterns and solving sequence-based problems effectively.
Q13. Can the common difference of an AP be zero? Explain.
Answer:
Yes, the common difference of an Arithmetic Progression can be zero. When d = 0, every term remains the same because no value is added or subtracted. For example:
5, 5, 5, 5, 5, …
Here,
d = 5 – 5 = 0
Since the difference between consecutive terms is constant, it satisfies the definition of an Arithmetic Progression. Such APs are called constant sequences. Even though the terms do not change, they still form a valid arithmetic progression because the common difference remains fixed throughout the sequence.
Q14. Find the 12th term of the AP: –3, 1, 5, 9, …
Answer:
Given:
a = –3
d = 1 – (–3) = 4
n = 12
Using:
an = a + (n – 1)d
a12 = –3 + (12 – 1) × 4
= –3 + 44
= 41
Therefore, the 12th term is 41. Arithmetic Progressions may contain negative numbers, positive numbers, or both. The same formula applies regardless of the sign of the terms, making it a universal tool for solving AP problems.
Q15. Why is the nth term formula important?
Answer:
The nth term formula is important because it enables us to find any term in an Arithmetic Progression directly. Without this formula, we would need to write all preceding terms, which becomes difficult for large values of n. The formula:
an = a + (n – 1)d
saves time and effort. It is useful in mathematical calculations, competitive examinations, and real-life situations involving regular increases or decreases. The formula also helps determine whether a given number belongs to an AP and assists in solving advanced sequence and series problems efficiently.
Q16. Find the 18th term of the AP: 11, 15, 19, 23, …
Answer:
Given:
a = 11
d = 15 – 11 = 4
n = 18
Using:
an = a + (n – 1)d
a18 = 11 + (18 – 1) × 4
= 11 + 68
= 79
Therefore, the 18th term is 79. The nth term formula is an essential tool in arithmetic progressions. It helps find distant terms quickly and accurately without constructing the entire sequence.
Q17. Find the first term if the common difference is 7 and the 6th term is 38.
Answer:
Given:
d = 7
a6 = 38
Using:
an = a + (n – 1)d
38 = a + (6 – 1) × 7
38 = a + 35
a = 3
Therefore, the first term is 3. Once the first term and common difference are known, the entire Arithmetic Progression can be generated. This type of problem frequently appears in CBSE examinations.
Q18. Find the common difference of the AP: –10, –6, –2, 2, …
Answer:
To find the common difference:
d = –6 – (–10)
= –6 + 10
= 4
Checking other terms:
–2 – (–6) = 4
2 – (–2) = 4
Since the difference remains constant, the sequence is an Arithmetic Progression. Therefore, the common difference is 4. Positive common differences indicate increasing sequences, while negative common differences indicate decreasing sequences.
Q19. Write an AP whose first term is 20 and common difference is –3.
Answer:
Given:
a = 20
d = –3
Each successive term is obtained by subtracting 3 from the previous term.
First term = 20
Second term = 17
Third term = 14
Fourth term = 11
Fifth term = 8
Thus, the AP is:
20, 17, 14, 11, 8, …
Since the common difference is negative, the sequence decreases regularly. Such APs are useful in representing situations involving constant reductions, such as depreciation or decreasing quantities over time.
Q20. State two real-life applications of Arithmetic Progressions.
Answer:
Arithmetic Progressions have many practical applications in everyday life. One application is in salary increments, where an employee may receive a fixed annual increase in salary. Another application is in stadium seating arrangements, where each row may contain a fixed number of seats more than the previous row. APs are also used in savings plans, construction designs, and population studies involving regular changes. Because the increase or decrease remains constant, arithmetic progressions provide a simple mathematical model for analyzing and predicting patterns. Their wide applicability makes them an important topic in mathematics and real-world problem-solving.