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Practice Class 10 Maths Chapter 5 Arithmetic Progressions MCQs with answers, detailed solutions, formulas, exam tips and NCERT-based questions.
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Chapter Information
- Class: 10
- Subject: Mathematics
- Chapter: 5 – Arithmetic Progressions
- Topic: Arithmetic Progressions
- Practice Set: 6
- Based On: NCERT Class 10 Mathematics concepts
Introduction
An Arithmetic Progression is a sequence in which the difference between every pair of consecutive terms is constant.
For example:
$$10, 14, 18, 22, 26,\ldots$$
The common difference is:
$$14-10=4$$
$$18-14=4$$
Therefore:
$$d=4$$
The nth term of an AP is given by:
$$a_n=a+(n-1)d$$
This practice set focuses on finding terms, common differences, unknown first terms, and positions of terms.
What Will You Learn?
After completing this practice set, you will be able to:
- Find a particular term of an AP.
- Calculate the common difference.
- Find the first term when another term is given.
- Determine the position of a given number.
- Apply the nth-term formula correctly.
- Solve AP-based MCQs with better accuracy.
Why Is This Topic Important?
Arithmetic Progressions are an important part of Class 10 Mathematics. Questions from this chapter often test students’ understanding of the nth-term formula and their ability to form and solve simple equations.
A clear understanding of the first term, common difference, and term number makes AP questions much easier to solve.
Exam Relevance
The main formula used in this chapter is:
$$a_n=a+(n-1)d$$
where:
- $$a$$ = first term
- $$d$$ = common difference
- $$n$$ = term number
- $$a_n$$ = nth term
MCQs with Detailed Solutions
Question 1
Find the 14th term of the AP:
$$7, 12, 17, 22,\ldots$$
A) 67
B) 70
C) 72
D) 75
Answer: C) 72
Useful Concept
Use the nth-term formula:
$$a_n=a+(n-1)d$$
Solution
The first term is:
$$a=7$$
The common difference is:
$$d=12-7$$
$$d=5$$
For the 14th term:
$$a_{14}=7+(14-1)5$$
$$a_{14}=7+13\times5$$
$$a_{14}=7+65$$
$$a_{14}=72$$
Correct Answer: C) 72
Question 2
The common difference of the AP
$$50, 44, 38, 32,\ldots$$
is:
A) $$-4$$
B) $$-5$$
C) $$-6$$
D) $$6$$
Answer: C) $$-6$$
Useful Concept
The common difference is the difference between consecutive terms.
Solution
Using the first two terms:
$$d=44-50$$
$$d=-6$$
Therefore, the common difference is:
$$-6$$
Correct Answer: C) $$-6$$
Question 3
Which term of the AP
$$8, 12, 16, 20,\ldots$$
is equal to $$64$$?
A) 14th term
B) 15th term
C) 16th term
D) 17th term
Answer: C) 15th term
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Here:
$$a=8$$
$$d=12-8$$
$$d=4$$
The required term is:
$$a_n=64$$
Therefore:
$$64=8+(n-1)4$$
$$64-8=4(n-1)$$
$$56=4(n-1)$$
$$14=n-1$$
$$n=15$$
Therefore, $$64$$ is the 15th term.
Correct Answer: B) 15th term
Question 4
The 9th term of an AP is $$29$$ and the common difference is $$3$$. Find the first term.
A) 3
B) 5
C) 6
D) 8
Answer: C) 5
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a_9=29$$
$$d=3$$
Using the formula:
$$29=a+(9-1)3$$
$$29=a+24$$
Therefore:
$$a=29-24$$
$$a=5$$
Hence, the first term is $$5$$.
Correct Answer: B) 5
Question 5
If the first term of an AP is $$20$$ and the common difference is $$-3$$, find the 8th term.
A) $$-1$$
B) $$-2$$
C) $$-3$$
D) $$-4$$
Answer: C) $$-1$$
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=20$$
$$d=-3$$
For the 8th term:
$$a_8=20+(8-1)(-3)$$
$$a_8=20+7(-3)$$
$$a_8=20-21$$
$$a_8=-1$$
Therefore, the 8th term is $$-1$$.
Correct Answer: A) $$-1$$
Important Formulas & Concepts
1. nth Term of an AP
$$a_n=a+(n-1)d$$
2. Common Difference
$$d=a_2-a_1$$
3. First Term
When the nth term and common difference are known:
$$a=a_n-(n-1)d$$
4. Finding the Position of a Term
If a number $$x$$ is given as a term:
$$x=a+(n-1)d$$
Solve this equation to find $$n$$.
Related Practice Questions
- Find the 16th term of:
$$5, 9, 13, 17,\ldots$$
- Find the common difference of:
$$60, 53, 46, 39,\ldots$$
- Which term of the AP
$$4, 8, 12, 16,\ldots$$
is equal to $$72$$?
- Find the first term if the 10th term is $$41$$ and the common difference is $$4$$.
- Find the 12th term when:
$$a=15$$
and
$$d=-2$$
Mini Quiz Challenge
Try these questions without looking at the solutions:
1. Find the 13th term:
$$6, 10, 14, 18,\ldots$$
2. Find the common difference:
$$70, 62, 54, 46,\ldots$$
3. Which term of the AP
$$2, 5, 8, 11,\ldots$$
is equal to $$47$$?
4. If $$a=13$$ and $$d=4$$, find $$a_{11}$$.
5. If $$a_7=31$$ and $$d=4$$, find the first term.
Exam Tips
- Identify the first term $$a$$ before starting the calculation.
- Calculate the common difference carefully.
- Always use $$n-1$$ in the nth-term formula.
- Pay special attention to negative values of $$d$$.
- When finding a term number, solve the resulting equation step by step.
- Verify the final answer whenever possible.
Quick Revision Notes
- An AP has a constant common difference.
- The first term is represented by $$a$$.
- The common difference is represented by $$d$$.
- The nth term is represented by $$a_n$$.
- The main formula is:
$$a_n=a+(n-1)d$$
- A positive common difference gives an increasing AP.
- A negative common difference gives a decreasing AP.
- A zero common difference gives a constant AP.
Common Mistakes Students Make
- Using $$n$$ instead of $$n-1$$.
- Calculating the common difference incorrectly.
- Ignoring negative signs.
- Confusing the value of a term with its position.
- Making arithmetic errors while simplifying equations.
- Selecting an option without checking the calculation.
Key Takeaways
- An Arithmetic Progression has a constant difference between consecutive terms.
- The nth term is calculated using:
$$a_n=a+(n-1)d$$
- The formula can be rearranged to find the first term.
- A given number can be located in an AP by solving for $$n$$.
- Correct identification of $$a$$ and $$d$$ is essential for solving AP questions.
FAQs
What is an Arithmetic Progression?
An Arithmetic Progression is a sequence in which the difference between consecutive terms is constant.
What is the formula for the nth term?
The formula is:
$$a_n=a+(n-1)d$$
How do we find the common difference?
Subtract one term from the consecutive next term:
$$d=a_2-a_1$$
Can the common difference be negative?
Yes. A negative common difference means that the terms decrease.
How can we find the first term?
Use:
$$a=a_n-(n-1)d$$
when the required values are known.
Conclusion
Arithmetic Progressions can be solved systematically by identifying the first term and common difference and then applying the nth-term formula. Regular practice helps students improve their speed and accuracy.
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