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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: 3
- Difficulty Level: Easy to Moderate
- Based On: NCERT Class 10 Mathematics Syllabus
Introduction
Arithmetic Progressions help students understand sequences in which consecutive terms have a constant difference. The concepts of the first term, common difference and nth term are essential for solving different types of AP questions.
In this practice set, the questions focus on determining unknown terms, comparing terms and applying the nth-term formula in different situations.
What You Will Learn?
After completing this practice set, you will be able to:
- Find unknown terms of an AP.
- Use the nth-term formula effectively.
- Determine the common difference from given terms.
- Find relationships between different terms of an AP.
- Solve numerical AP problems accurately.
Why This Topic Is Important?
Arithmetic Progressions are an important part of Class 10 Mathematics. Questions from AP may require students to find unknown values by forming equations using the nth-term formula.
Practising different forms of AP questions improves calculation skills and helps students prepare for examinations.
Exam Relevance
Important concepts covered in this practice set include:
- nth term
- First term
- Common difference
- Unknown terms
- Relationship between terms
- Formation of equations
MCQs with Detailed Solutions
Q1. If the first term of an AP is 8 and the common difference is 5, what is the 9th term?
A) 43
B) 48
C) 53
D) 58
Answer: C) 48
Useful Concept
The nth term of an AP is:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=8$$
$$d=5$$
$$n=9$$
Using the formula:
$$a_9=8+(9-1)5$$
$$a_9=8+8\times5$$
$$a_9=8+40$$
$$a_9=48$$
Therefore, the 9th term is 48.
Correct Answer: B
Q2. The 4th term of an AP is 16 and the common difference is 3. What is the 10th term?
A) 31
B) 32
C) 34
D) 35
Answer: C) 34
Useful Concept
First find the first term using:
$$a_n=a+(n-1)d$$
Then calculate the required term.
Solution
Given:
$$a_4=16$$
$$d=3$$
Using the nth-term formula:
$$a_4=a+(4-1)3$$
$$16=a+3\times3$$
$$16=a+9$$
Therefore:
$$a=16-9$$
$$a=7$$
Now find the 10th term:
$$a_{10}=7+(10-1)3$$
$$a_{10}=7+9\times3$$
$$a_{10}=7+27$$
$$a_{10}=34$$
Therefore, the 10th term is 34.
Correct Answer: C
Q3. In an AP, the 6th term is 22 and the 10th term is 34. What is the common difference?
A) 2
B) 3
C) 4
D) 5
Answer: B) 3
Useful Concept
Use the nth-term formula for both given terms and subtract the resulting equations.
Solution
For the 6th term:
$$a_6=a+(6-1)d$$
Therefore:
$$a+5d=22$$
For the 10th term:
$$a_{10}=a+(10-1)d$$
Therefore:
$$a+9d=34$$
Subtract the first equation from the second:
$$a+9d-(a+5d)=34-22$$
$$4d=12$$
Therefore:
$$d=3$$
Hence, the common difference is 3.
Correct Answer: B
Q4. Which term of the AP 20, 16, 12, 8, … is -16?
A) 8th term
B) 9th term
C) 10th term
D) 11th term
Answer: C) 10th term
Useful Concept
The common difference can be negative. Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=20$$
$$d=16-20$$
$$d=-4$$
The required term is:
$$a_n=-16$$
Using the formula:
$$-16=20+(n-1)(-4)$$
Therefore:
$$-16=20-4(n-1)$$
Subtract 20 from both sides:
$$-16-20=-4(n-1)$$
$$-36=-4(n-1)$$
Divide by -4:
$$9=n-1$$
Therefore:
$$n=10$$
Hence, -16 is the 10th term of the AP.
Correct Answer: C
Q5. If the 3rd term of an AP is 12 and the 8th term is 27, what is the 15th term?
A) 45
B) 48
C) 51
D) 54
Answer: C) 48
Useful Concept
First find the common difference using the two given terms. Then find the first term and use the nth-term formula.
Solution
For the 3rd term:
$$a_3=a+(3-1)d$$
Therefore:
$$a+2d=12$$
For the 8th term:
$$a_8=a+(8-1)d$$
Therefore:
$$a+7d=27$$
Subtract the first equation from the second:
$$a+7d-(a+2d)=27-12$$
$$5d=15$$
Therefore:
$$d=3$$
Now use:
$$a+2d=12$$
Substitute $$d=3$$:
$$a+2(3)=12$$
$$a+6=12$$
Therefore:
$$a=6$$
Now find the 15th term:
$$a_{15}=6+(15-1)3$$
$$a_{15}=6+14\times3$$
$$a_{15}=6+42$$
$$a_{15}=48$$
Therefore, the 15th term is 48.
Correct Answer: B
Important Formulas & Concepts
1. nth Term of an AP
$$a_n=a+(n-1)d$$
2. First Term
The first term can be found using:
$$a=a_n-(n-1)d$$
3. Common Difference
For consecutive terms:
$$d=a_{n+1}-a_n$$
4. Negative Common Difference
An AP can have a negative common difference.
For example:
$$20,16,12,8,\ldots$$
Here:
$$d=16-20$$
$$d=-4$$
5. Difference Between Two Terms
If two terms are known, the difference between them can help determine the common difference.
For example:
$$a_m-a_n=(m-n)d$$
Related Practice Questions
Try solving these questions yourself:
- Find the 11th term of the AP 9, 13, 17, 21, …
- If the 5th term of an AP is 19 and the common difference is 4, find the first term.
- The 7th term of an AP is 25 and the 12th term is 40. Find the common difference.
- Which term of the AP 30, 26, 22, 18, … is -6?
- If the 4th term of an AP is 13 and the 9th term is 28, find the 14th term.
Mini Quiz Challenge
Test your understanding:
1. Find the 12th term of the AP 7, 10, 13, 16, …
2. Find the common difference if the 5th term is 18 and the 9th term is 30.
3. Which term of the AP 2, 6, 10, 14, … is 42?
4. If the 3rd term of an AP is 10 and the common difference is 4, find the first term.
5. Find the 20th term of the AP 25, 22, 19, 16, …
Exam Tips
- Write the given terms and values clearly before starting.
- Use the nth-term formula carefully.
- Remember that the common difference can be positive or negative.
- When two terms are given, form two equations and subtract them.
- Check the sign of the common difference before calculating.
- Substitute the final values back into the formula to verify your answer.
- Avoid calculation mistakes when working with negative numbers.
Quick Revision Notes
- The nth term of an AP is:
$$a_n=a+(n-1)d$$
- The first term can be found using:
$$a=a_n-(n-1)d$$
- The common difference is:
$$d=a_{n+1}-a_n$$
- The common difference may be positive, zero or negative.
- If the terms decrease, the common difference is negative.
- Two known terms can be used to form equations and determine unknown values.
Common Mistakes Students Make
- Forgetting the negative sign when the AP is decreasing.
- Using the wrong term number in the nth-term formula.
- Subtracting equations incorrectly.
- Confusing the first term with the common difference.
- Making sign errors while solving equations.
- Not verifying the final answer with the given terms.
Key Takeaways
- The nth-term formula is useful for finding any term of an AP.
- A negative common difference represents a decreasing AP.
- Two known terms can help determine the common difference.
- Equations can be formed using the nth-term formula.
- Careful handling of positive and negative values is important while solving AP questions.
FAQs
Q1. Can the common difference of an AP be negative?
Yes. For example:
$$20,16,12,8,\ldots$$
has:
$$d=-4$$
Q2. How do we find the nth term of an AP?
Use:
$$a_n=a+(n-1)d$$
Q3. How can two given terms help find the common difference?
The nth-term formula can be applied to both terms. Subtracting the resulting equations can give the value of $$d$$.
Q4. What happens when the common difference is zero?
If:
$$d=0$$
then all terms of the AP are equal.
Q5. What does a positive common difference indicate?
A positive common difference means that the terms increase as the sequence progresses.
Conclusion
Arithmetic Progressions can contain increasing, decreasing or constant terms depending on the value of the common difference. Understanding how to use the nth-term formula with different given conditions is essential for solving AP problems.
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