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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: 5
- Based On: NCERT Class 10 Mathematics concepts
Introduction
An Arithmetic Progression is a sequence in which the difference between any two consecutive terms remains constant.
For example:
$$6, 10, 14, 18, 22,\ldots$$
The common difference is:
$$10-6=4$$
$$14-10=4$$
Therefore, the common difference is:
$$d=4$$
The nth term of an AP can be found using:
$$a_n=a+(n-1)d$$
This practice set provides five MCQs covering important concepts of Arithmetic Progressions.
What Will You Learn?
After completing this practice set, you will be able to:
- Find missing terms in an AP.
- Calculate the nth term.
- Find the common difference.
- Determine the position of a particular term.
- Find the first term when other information is given.
- Apply AP formulas accurately.
Why Is This Topic Important?
Arithmetic Progressions are an important part of Class 10 Mathematics. Questions based on APs can involve direct formula application as well as algebraic reasoning.
Understanding the relationship between the first term, common difference, term number, and nth term helps students solve examination questions efficiently.
Exam Relevance
The most important formula for finding a particular term is:
$$a_n=a+(n-1)d$$
Here:
- $$a$$ = first term
- $$d$$ = common difference
- $$n$$ = term number
- $$a_n$$ = nth term
MCQs with Detailed Solutions
Question 1
What is the 15th term of the AP?
$$4, 7, 10, 13,\ldots$$
A) 43
B) 44
C) 46
D) 48
Answer: C) 46
Useful Concept
Use the nth-term formula:
$$a_n=a+(n-1)d$$
Solution
The first term is:
$$a=4$$
The common difference is:
$$d=7-4$$
$$d=3$$
For the 15th term:
$$a_{15}=4+(15-1)3$$
$$a_{15}=4+14\times3$$
$$a_{15}=4+42$$
$$a_{15}=46$$
Correct Answer: C) 46
Question 2
Which of the following is the common difference of the AP?
$$32, 27, 22, 17,\ldots$$
A) $$-3$$
B) $$-4$$
C) $$-5$$
D) $$5$$
**Answer: C) $$-5$$
Useful Concept
The common difference is the difference between consecutive terms.
Solution
Using the first two terms:
$$d=27-32$$
$$d=-5$$
Therefore, the common difference is:
$$-5$$
**Correct Answer: C) $$-5$$
Question 3
Which term of the AP
$$9, 13, 17, 21,\ldots$$
is equal to $$57$$?
A) 11th term
B) 12th term
C) 13th term
D) 14th term
Answer: C) 13th term
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Here:
$$a=9$$
$$d=13-9$$
$$d=4$$
The required term is:
$$a_n=57$$
Therefore:
$$57=9+(n-1)4$$
$$57-9=4(n-1)$$
$$48=4(n-1)$$
$$12=n-1$$
$$n=13$$
Therefore, $$57$$ is the 13th term.
Correct Answer: C) 13th term
Question 4
If the first term of an AP is $$15$$ and the common difference is $$4$$, which term is $$51$$?
A) 9th term
B) 10th term
C) 11th term
D) 12th term
Answer: C) 10th term
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a=15$$
$$d=4$$
The required term is:
$$a_n=51$$
Therefore:
$$51=15+(n-1)4$$
$$51-15=4(n-1)$$
$$36=4(n-1)$$
$$9=n-1$$
$$n=10$$
Therefore, $$51$$ is the 10th term.
Correct Answer: B) 10th term
Question 5
The 7th term of an AP is $$25$$ and its common difference is $$3$$. Find the first term.
A) 5
B) 6
C) 7
D) 8
Answer: C) 7
Useful Concept
Use:
$$a_n=a+(n-1)d$$
Solution
Given:
$$a_7=25$$
$$d=3$$
Using the formula:
$$25=a+(7-1)3$$
$$25=a+18$$
Therefore:
$$a=25-18$$
$$a=7$$
Hence, the first term is $$7$$.
Correct Answer: C) 7
Important Formulas & Concepts
1. nth Term of an AP
$$a_n=a+(n-1)d$$
2. Common Difference
$$d=a_2-a_1$$
3. Finding the First Term
$$a=a_n-(n-1)d$$
4. Finding the Position of a Term
If a number $$x$$ is a term of an AP:
$$x=a+(n-1)d$$
Solve this equation to find $$n$$.
Related Practice Questions
- Find the 18th term of:
$$2, 6, 10, 14,\ldots$$
- Find the common difference of:
$$40, 34, 28, 22,\ldots$$
- Which term of the AP
$$5, 9, 13, 17,\ldots$$
is equal to $$61$$?
- Find the first term if the 9th term is $$35$$ and the common difference is $$4$$.
- Determine whether $$50$$ is a term of the AP:
$$2, 6, 10, 14,\ldots$$
Mini Quiz Challenge
Try these without looking at the solutions:
1. Find the 11th term:
$$8, 12, 16, 20,\ldots$$
2. Find the common difference:
$$45, 39, 33, 27,\ldots$$
3. Which term of the AP
$$3, 8, 13, 18,\ldots$$
is equal to $$48$$?
4. If $$a=11$$ and $$d=2$$, find $$a_{14}$$.
5. If $$a_8=31$$ and $$d=3$$, find the first term.
Exam Tips
- Identify $$a$$ and $$d$$ before applying the formula.
- Remember that the formula contains $$n-1$$.
- Be careful with negative common differences.
- Write the equation clearly when finding the position of a term.
- Check your final answer using the original AP whenever possible.
- Read all four options carefully before selecting the answer.
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:
$$a_n=a+(n-1)d$$
- Positive $$d$$ produces an increasing AP.
- Negative $$d$$ produces a decreasing AP.
- Zero $$d$$ produces a constant AP.
Common Mistakes Students Make
- Writing $$n$$ instead of $$n-1$$.
- Calculating the common difference in the wrong order.
- Forgetting the negative sign in a decreasing AP.
- Confusing the term number with the term value.
- Making arithmetic mistakes while simplifying the equation.
- Choosing an answer without verifying the calculation.
Key Takeaways
- Arithmetic Progressions follow a constant difference between consecutive terms.
- The nth term is calculated using:
$$a_n=a+(n-1)d$$
- The same formula can be rearranged to find the first term.
- A given number can be checked as a term by solving for $$n$$.
- Careful substitution and calculation are essential for accurate answers.
FAQs
What is an Arithmetic Progression?
An Arithmetic Progression is a sequence in which the difference between consecutive terms is constant.
What is the nth-term formula?
The nth-term formula is:
$$a_n=a+(n-1)d$$
How is the common difference found?
Subtract a term from the next consecutive term:
$$d=a_2-a_1$$
Can an AP have a negative common difference?
Yes. A negative common difference means that the terms decrease.
How do we find the first term?
Use:
$$a=a_n-(n-1)d$$
when the nth term, term number, and common difference are known.
Conclusion
Arithmetic Progressions can be solved efficiently by identifying the first term and common difference and then applying the correct formula. Regular practice helps students improve both speed and accuracy.
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