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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: 4
- Based On: NCERT Class 10 Mathematics concepts
Introduction
Arithmetic Progressions are an important topic in Class 10 Mathematics. An arithmetic progression is a sequence in which the difference between consecutive terms remains constant.
For example:
$$4, 9, 14, 19, 24,\ldots$$
Here,
$$9-4=5$$
$$14-9=5$$
$$19-14=5$$
Therefore, the common difference is constant.
In this practice set, students will solve MCQs related to the general term, common difference, finding unknown terms, and identifying specific terms of an arithmetic progression.
What Will You Learn?
After completing this practice set, you will be able to:
- Identify the common difference of an AP.
- Find the first term of an AP.
- Find a particular term using the general term formula.
- Determine whether a number is a term of an AP.
- Apply the formula for the nth term correctly.
- Solve AP-based MCQs efficiently.
Why Is This Topic Important?
Arithmetic Progressions are frequently used in Class 10 Mathematics examinations. Questions may require students to find a particular term, identify the common difference, or determine the position of a given number in an AP.
A strong understanding of the nth-term formula helps students solve these questions accurately and quickly.
Exam Relevance
Students should remember the general term formula:
$$a_n=a+(n-1)d$$
where:
- $$a$$ = first term
- $$d$$ = common difference
- $$n$$ = number of terms
- $$a_n$$ = nth term
MCQs with Detailed Solutions
Question 1
Find the 12th term of the AP:
$$5, 8, 11, 14,\ldots$$
A) 35
B) 36
C) 38
D) 40
Answer: C) 38
Useful Concept
The nth term of an AP is:
$$a_n=a+(n-1)d$$
Solution
Here,
$$a=5$$
The common difference is:
$$d=8-5$$
$$d=3$$
For the 12th term:
$$a_{12}=5+(12-1)3$$
$$a_{12}=5+11\times3$$
$$a_{12}=5+33$$
$$a_{12}=38$$
Correct Answer: C) 38
Question 2
The common difference of the AP
$$18, 13, 8, 3,\ldots$$
is:
A) $$-3$$
B) $$-4$$
C) $$-5$$
D) $$5$$
Answer: C) $$-5$$
Useful Concept
The common difference is obtained by subtracting a term from the next term:
$$d=a_2-a_1$$
Solution
Using the first two terms:
$$d=13-18$$
$$d=-5$$
Therefore, the common difference is:
$$-5$$
Correct Answer: C) $$-5$$
Question 3
Which term of the AP
$$7, 11, 15, 19,\ldots$$
is equal to $$51$$?
A) 10th term
B) 11th term
C) 12th term
D) 13th term
Answer: C) 12th term
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Here,
$$a=7$$
$$d=11-7$$
$$d=4$$
We need:
$$a_n=51$$
Therefore,
$$51=7+(n-1)4$$
$$51-7=4(n-1)$$
$$44=4(n-1)$$
$$11=n-1$$
$$n=12$$
Therefore, $$51$$ is the 12th term.
Correct Answer: C) 12th term
Question 4
If the first term of an AP is $$12$$ and its common difference is $$-2$$, then which term is equal to $$-18$$?
A) 14th term
B) 15th term
C) 16th term
D) 17th term
Answer: C) 16th term
Useful Concept
The nth term formula is:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=12$$
$$d=-2$$
The required term is:
$$a_n=-18$$
Therefore,
$$-18=12+(n-1)(-2)$$
$$-18=12-2(n-1)$$
Subtracting $$12$$ from both sides:
$$-30=-2(n-1)$$
Dividing by $$-2$$:
$$15=n-1$$
Therefore,
$$n=16$$
Hence, $$-18$$ is the 16th term.
Correct Answer: C) 16th term
Question 5
The 5th term of an AP is $$17$$ and its common difference is $$3$$. What is the first term?
A) 5
B) 6
C) 7
D) 8
Answer: C) 5
Useful Concept
Use the nth term formula:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a_5=17$$
$$d=3$$
Using the formula:
$$17=a+(5-1)3$$
$$17=a+12$$
Therefore,
$$a=17-12$$
$$a=5$$
Hence, the first term is $$5$$.
Correct Answer: A) 5
Important Formulas & Concepts
1. General Term of an AP
$$a_n=a+(n-1)d$$
2. Common Difference
$$d=a_2-a_1$$
or
$$d=a_3-a_2$$
3. First Term
If the nth term, number of terms, and common difference are known:
$$a=a_n-(n-1)d$$
4. Finding the Position of a Term
If a particular value $$x$$ is given:
$$x=a+(n-1)d$$
Then solve the equation to find $$n$$.
Related Practice Questions
- Find the 15th term of the AP:
$$3, 7, 11, 15,\ldots$$
- Find the common difference of:
$$25, 20, 15, 10,\ldots$$
- Which term of the AP
$$6, 10, 14, 18,\ldots$$
is equal to $$62$$?
- Find the first term if the 8th term is $$29$$ and the common difference is $$4$$.
- Determine whether $$45$$ is a term of the AP:
$$5, 9, 13, 17,\ldots$$
Mini Quiz Challenge
Try these questions without looking at the solutions:
1. Find the 9th term of:
$$2, 6, 10, 14,\ldots$$
2. Find the common difference of:
$$30, 24, 18, 12,\ldots$$
3. Which term of the AP
$$4, 7, 10, 13,\ldots$$
is equal to $$40$$?
4. If $$a=9$$ and $$d=5$$, find $$a_{10}$$.
5. If $$a_6=32$$ and $$d=4$$, find the first term.
Exam Tips
- First identify the first term $$a$$.
- Find the common difference $$d$$ carefully.
- Use $$a_n=a+(n-1)d$$ for finding the nth term.
- Pay special attention to negative common differences.
- When finding the position of a term, solve the equation carefully.
- Always substitute your answer back into the formula when possible.
- Check the final answer against the given options.
Quick Revision Notes
- An AP has a constant difference between consecutive terms.
- The first term is represented by $$a$$.
- The common difference is represented by $$d$$.
- The nth term is represented by $$a_n$$.
- The nth term formula is:
$$a_n=a+(n-1)d$$
- A positive $$d$$ indicates an increasing AP.
- A negative $$d$$ indicates a decreasing AP.
- A zero value of $$d$$ gives a constant sequence.
Common Mistakes Students Make
- Using $$n$$ instead of $$n-1$$ in the nth-term formula.
- Calculating the common difference with the wrong order.
- Ignoring the negative sign in a decreasing AP.
- Confusing the term number with the value of the term.
- Selecting an option without checking the calculation.
- Forgetting to identify the first term correctly.
Key Takeaways
- Arithmetic Progression is based on a constant common difference.
- The most important formula is:
$$a_n=a+(n-1)d$$
- The common difference can be positive, negative, or zero.
- The nth-term formula can also be rearranged to find the first term or the position of a term.
- Careful substitution helps avoid calculation mistakes.
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 of an AP?
The formula is:
$$a_n=a+(n-1)d$$
What does $$d$$ represent in an AP?
$$d$$ represents the common difference between consecutive terms.
How can we find the first term?
If the nth term, number of terms, and common difference are known, use:
$$a=a_n-(n-1)d$$
Can the common difference be negative?
Yes. A negative common difference represents a decreasing arithmetic progression.
Conclusion
Arithmetic Progressions become easier when students understand the common difference and correctly apply the nth-term formula. Regular practice with different types of AP questions can improve accuracy and exam performance.
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