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Practice Class 10 Maths Chapter 5 Arithmetic Progressions MCQs with answers, formulas and detailed solutions based on the NCERT syllabus.
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Chapter Information
- Subject: Mathematics
- Class: 10
- Chapter: Arithmetic Progressions
- Question Type: MCQs
- Practice Set: 2
- Difficulty Level: Easy to Moderate
- Based On: NCERT Class 10 Mathematics Syllabus
Introduction
Arithmetic Progressions are an important part of Class 10 Mathematics. In an AP, the difference between consecutive terms remains constant. Understanding the relationship between the first term, common difference, number of terms and other terms helps students solve different types of problems.
This practice set focuses on finding the first term, identifying the common difference from given information, finding specific terms and solving basic AP problems.
What You Will Learn?
After completing this practice set, you will be able to:
- Find the first term of an AP.
- Calculate the common difference using given terms.
- Find a particular term of an AP.
- Use the nth-term formula correctly.
- Solve basic problems involving Arithmetic Progressions.
- Apply AP concepts accurately in examinations.
Why This Topic Is Important?
Arithmetic Progressions are frequently used in Class 10 Mathematics questions. Students need to understand how different terms of an AP are connected through the common difference.
A strong understanding of these concepts helps students solve both MCQs and descriptive questions with greater accuracy.
Exam Relevance
Important areas covered in this practice set include:
- First term
- Common difference
- nth term
- Position of a term
- Relationship between terms
- Basic AP calculations
MCQs with Detailed Solutions
Q1. The first term of an AP is 6 and its common difference is 4. What is its 7th term?
A) 26
B) 28
C) 30
D) 32
Answer: C) 30
Useful Concept
The nth term of an AP is:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=6$$
$$d=4$$
$$n=7$$
Using the nth-term formula:
$$a_7=6+(7-1)(4)$$
$$a_7=6+6\times4$$
$$a_7=6+24$$
$$a_7=30$$
Therefore, the 7th term is 30.
Correct Answer: C
Q2. If the 5th term of an AP is 18 and the common difference is 3, what is its first term?
A) 3
B) 6
C) 9
D) 12
Answer: B) 6
Useful Concept
The nth term of an AP is:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a_5=18$$
$$d=3$$
and:
$$n=5$$
Using the formula:
$$a_5=a+(5-1)3$$
$$18=a+4\times3$$
$$18=a+12$$
Subtract 12 from both sides:
$$18-12=a$$
$$a=6$$
Therefore, the first term is 6.
Correct Answer: B
Q3. The 3rd term of an AP is 11 and the 7th term is 23. What is the common difference?
A) 2
B) 3
C) 4
D) 5
Answer: B) 3
Useful Concept
For an AP:
$$a_n=a+(n-1)d$$
We can form equations using the given terms.
Solution
For the 3rd term:
$$a_3=a+(3-1)d$$
Therefore:
$$a+2d=11$$
For the 7th term:
$$a_7=a+(7-1)d$$
Therefore:
$$a+6d=23$$
Subtract the first equation from the second:
$$a+6d-(a+2d)=23-11$$
$$4d=12$$
Divide by 4:
$$d=3$$
Therefore, the common difference is 3.
Correct Answer: B
Q4. Which term of the AP 12, 17, 22, 27, … is 57?
A) 8th term
B) 9th term
C) 10th term
D) 11th term
Answer: C) 10th term
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=12$$
$$d=17-12$$
$$d=5$$
The required term is:
$$a_n=57$$
Using the formula:
$$57=12+(n-1)5$$
Subtract 12:
$$57-12=5(n-1)$$
$$45=5(n-1)$$
Divide by 5:
$$\frac{45}{5}=n-1$$
$$9=n-1$$
Therefore:
$$n=10$$
Hence, 57 is the 10th term of the AP.
Correct Answer: C
Q5. If the 8th term of an AP is 24 and the common difference is 2, what is the 12th term?
A) 30
B) 31
C) 32
D) 34
Answer: C) 32
Useful Concept
First find the first term using:
$$a_n=a+(n-1)d$$
Then use the nth-term formula to find the required term.
Solution
Given:
$$a_8=24$$
$$d=2$$
Using the nth-term formula:
$$a_8=a+(8-1)2$$
$$24=a+7\times2$$
$$24=a+14$$
Therefore:
$$a=24-14$$
$$a=10$$
Now find the 12th term:
$$a_{12}=a+(12-1)d$$
$$a_{12}=10+11\times2$$
$$a_{12}=10+22$$
$$a_{12}=32$$
Therefore, the 12th term is 32.
Correct Answer: C
Important Formulas & Concepts
1. nth Term of an AP
$$a_n=a+(n-1)d$$
Here, $$a$$ is the first term, $$d$$ is the common difference and $$n$$ represents the term number.
2. Finding the First Term
From:
$$a_n=a+(n-1)d$$
we can write:
$$a=a_n-(n-1)d$$
3. Finding the Common Difference
If two terms of an AP are known, equations can be formed using:
$$a_n=a+(n-1)d$$
and then solved to find $$d$$.
4. Difference Between Two Terms
For an AP:
$$a_m=a+(m-1)d$$
and:
$$a_n=a+(n-1)d$$
The difference between the terms can be used to determine the common difference.
5. Three Consecutive Terms
If three consecutive terms are represented by $$a$$, $$b$$ and $$c$$, then:
$$b-a=c-b$$
Therefore:
$$2b=a+c$$
Related Practice Questions
Try solving these questions yourself:
- Find the 9th term of the AP 4, 9, 14, 19, …
- If the 6th term of an AP is 25 and the common difference is 4, find the first term.
- The 4th term of an AP is 14 and the 8th term is 30. Find the common difference.
- Which term of the AP 7, 11, 15, 19, … is 51?
- If the 10th term of an AP is 35 and its common difference is 3, find the first term.
Mini Quiz Challenge
Test your understanding:
1. Find the 10th term of the AP 5, 8, 11, 14, …
2. If the first term is 9 and the common difference is 5, find the 6th term.
3. If the 4th term of an AP is 15 and the common difference is 2, find the first term.
4. Which term of the AP 3, 7, 11, 15, … is 39?
5. If the 6th term of an AP is 20 and the common difference is 3, find the 10th term.
Exam Tips
- Write the given values clearly before starting the calculation.
- Identify the first term and common difference carefully.
- Use the correct term number in the formula.
- When two terms are given, form equations carefully.
- Check the final answer by substituting it back into the AP formula.
- Avoid skipping calculation steps in numerical questions.
- Always remember the term formula:
$$a_n=a+(n-1)d$$
Quick Revision Notes
- First term of an AP is represented by $$a$$.
- Common difference is represented by $$d$$.
- nth term is:
$$a_n=a+(n-1)d$$
- First term can be found using:
$$a=a_n-(n-1)d$$
- For three consecutive terms:
$$2b=a+c$$
- The difference between consecutive terms of an AP remains constant.
Common Mistakes Students Make
- Using $$n$$ instead of $$(n-1)$$ in the nth-term formula.
- Confusing the first term with the term number.
- Using the wrong common difference.
- Forgetting to subtract the known term when finding the first term.
- Making sign errors while solving equations.
- Not checking whether the final answer satisfies the given conditions.
Key Takeaways
- The nth-term formula is one of the most important formulas in Arithmetic Progressions.
- The first term and common difference determine every term of an AP.
- Two known terms can be used to form equations and find the common difference.
- Careful substitution helps avoid calculation errors.
- Regular practice improves speed and accuracy in AP questions.
FAQs
Q1. What is the formula for the nth term of an AP?
The nth term is given by:
$$a_n=a+(n-1)d$$
Q2. How can the first term of an AP be found?
The first term can be found using:
$$a=a_n-(n-1)d$$
Q3. What does $$d$$ represent in an AP?
$$d$$ represents the common difference between consecutive terms.
Q4. Can two terms of an AP be used to find the common difference?
Yes. The nth-term formula can be used to form equations from the given terms and determine the common difference.
Q5. What is the relationship between three consecutive terms of an AP?
If the three consecutive terms are $$a$$, $$b$$ and $$c$$, then:
$$2b=a+c$$
Conclusion
Arithmetic Progressions become easier when students understand the relationship between the first term, common difference and term number. The nth-term formula can be used to find unknown terms and solve different types of AP problems.
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