Geometric Progression (GP) for Competitive Exam Success

Geometric Progression (GP)

Q. In a geometric progression, if the 3rd term is 27 and the common ratio is 3, what is the first term?

A. 3
B. 9
C. 1
D. 27

Solution

Let the first term be a. The 3rd term is a * r^2 = a * 3^2 = 9a. Setting 9a = 27 gives a = 3.

Correct Answer: B — 9

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Q. In a geometric progression, if the first term is 3 and the common ratio is 2, what is the 5th term?

A. 48
B. 24
C. 12
D. 6

Solution

The nth term of a GP is given by a * r^(n-1). Here, a = 3, r = 2, and n = 5. Thus, the 5th term = 3 * 2^(5-1) = 3 * 16 = 48.

Correct Answer: A — 48

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Q. In a geometric progression, if the first term is 4 and the common ratio is 1/2, what is the 6th term?

A. 0.25
B. 0.5
C. 1
D. 2

Solution

The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 4/32 = 0.125.

Correct Answer: A — 0.25

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Q. In a geometric progression, if the first term is 5 and the common ratio is 0.5, what is the sum of the first 4 terms?

A. 5
B. 10
C. 15
D. 20

Solution

The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_4 = 5(1 - 0.5^4) / (1 - 0.5) = 5(1 - 0.0625) / 0.5 = 5 * 0.9375 / 0.5 = 9.375.

Correct Answer: B — 10

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Q. In a geometric progression, if the first term is 5 and the last term is 80 with 4 terms in total, what is the common ratio?

A. 2
B. 3
C. 4
D. 5

Solution

The last term can be expressed as a * r^(n-1). Here, 80 = 5 * r^(4-1) = 5 * r^3. Thus, r^3 = 16, giving r = 2.

Correct Answer: A — 2

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Q. In a geometric progression, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?

A. 2
B. 3
C. 4
D. 5

Solution

Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.

Correct Answer: A — 2

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Q. In a geometric progression, if the first term is x and the common ratio is r, what is the expression for the sum of the first n terms?

A. x(1 - r^n)/(1 - r)
B. x(1 + r^n)/(1 + r)
C. xr^n/(1 - r)
D. xr^n/(1 + r)

Solution

The sum of the first n terms of a GP is given by S_n = a(1 - r^n)/(1 - r) for r ≠ 1.

Correct Answer: A — x(1 - r^n)/(1 - r)

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Q. In a geometric progression, if the first term is x and the common ratio is y, what is the expression for the 3rd term?

A. xy^2
B. x/y^2
C. x^2y
D. x^2/y

Solution

The 3rd term of a GP is given by a * r^(n-1). Here, it is x * y^(3-1) = xy^2.

Correct Answer: A — xy^2

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Q. In a GP, if the 3rd term is 27 and the 5th term is 243, what is the first term?

A. 3
B. 9
C. 1
D. 27

Solution

Let the first term be a and the common ratio be r. Then, 3rd term = ar^2 = 27 and 5th term = ar^4 = 243. Dividing gives r^2 = 9, so r = 3. Substituting back gives a = 3.

Correct Answer: B — 9

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Q. In a GP, if the first term is 10 and the common ratio is 0.5, what is the 6th term?

A. 0.625
B. 1.25
C. 2.5
D. 5

Solution

The 6th term is given by 10 * (0.5)^(6-1) = 10 * (0.5)^5 = 10 * 0.03125 = 0.3125.

Correct Answer: A — 0.625

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Q. In a GP, if the first term is 2 and the common ratio is -2, what is the 4th term?

A. 8
B. -8
C. 32
D. -32

Solution

The 4th term is given by 2 * (-2)^(4-1) = 2 * (-8) = -16.

Correct Answer: B — -8

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Q. In a GP, if the first term is 4 and the common ratio is 1/2, what is the 6th term?

A. 0.25
B. 0.5
C. 1
D. 2

Solution

The 6th term is given by a * r^(n-1) = 4 * (1/2)^(6-1) = 4 * (1/32) = 0.125, which is 0.25.

Correct Answer: A — 0.25

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Q. In a GP, if the first term is 5 and the common ratio is 1/2, what is the sum of the first four terms?

A. 15
B. 10
C. 12.5
D. 20

Solution

The first four terms are 5, 2.5, 1.25, and 0.625. Their sum is 5 + 2.5 + 1.25 + 0.625 = 9.375.

Correct Answer: C — 12.5

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Q. In a GP, if the first term is 5 and the last term is 80, and there are 4 terms in total, what is the common ratio?

A. 2
B. 3
C. 4
D. 5

Solution

Let the common ratio be r. The terms are 5, 5r, 5r^2, 5r^3. Setting 5r^3 = 80 gives r^3 = 16, thus r = 2.

Correct Answer: A — 2

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Q. In a GP, if the first term is 7 and the common ratio is 1/2, what is the 6th term?

A. 0.4375
B. 0.5
C. 1
D. 1.75

Solution

The 6th term is given by 7 * (1/2)^(6-1) = 7 * (1/32) = 0.4375.

Correct Answer: A — 0.4375

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Q. In a GP, if the first term is x and the common ratio is y, what is the expression for the 6th term?

A. xy^5
B. xy^6
C. x^6y
D. x^5y

Solution

The nth term of a GP is given by a * r^(n-1). Thus, the 6th term = x * y^(6-1) = xy^5.

Correct Answer: A — xy^5

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Q. What is the relationship between the first term and the common ratio if the sum of an infinite GP converges?

A. The first term must be zero.
B. The common ratio must be less than one in absolute value.
C. The first term must be greater than the common ratio.
D. The common ratio must be greater than one.

Solution

For the sum of an infinite GP to converge, the common ratio must be less than one in absolute value.

Correct Answer: B — The common ratio must be less than one in absolute value.

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Q. What is the sum of the first 5 terms of a GP where the first term is 2 and the common ratio is 3?

A. 242
B. 364
C. 486
D. 728

Solution

The sum of the first n terms of a GP is given by S_n = a(1 - r^n) / (1 - r). Here, S_5 = 2(1 - 3^5) / (1 - 3) = 2(1 - 243) / (-2) = 242.

Correct Answer: A — 242

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Q. Which of the following is NOT a characteristic of a geometric progression?

A. The ratio of any two consecutive terms is constant.
B. The product of the first and last terms equals the square of the middle term.
C. The sum of the terms can be negative.
D. The terms can be non-numeric.

Solution

In a geometric progression, the terms are numeric and follow a specific ratio; thus, non-numeric terms do not fit the definition.

Correct Answer: D — The terms can be non-numeric.

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Q. Which of the following is NOT a property of a geometric progression?

A. The product of the first and last terms equals the square of the geometric mean.
B. The sum of the terms can be negative.
C. The ratio of the last term to the first term is equal to the common ratio raised to the power of (n-1).
D. The terms can be non-numeric.

Solution

In a geometric progression, the terms are numeric and follow a specific ratio; thus, non-numeric terms do not form a valid GP.

Correct Answer: D — The terms can be non-numeric.

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Q. Which of the following is NOT a property of geometric progressions?

A. The product of the terms is equal to the square of the geometric mean.
B. The sum of the terms can be negative.
C. The common ratio can be zero.
D. The terms can be fractions.

Solution

In a geometric progression, the common ratio cannot be zero, as it would invalidate the progression.

Correct Answer: C — The common ratio can be zero.

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Q. Which of the following sequences is a geometric progression?

A. 1, 2, 4, 8
B. 1, 3, 6, 10
C. 2, 4, 8, 16
D. 1, 1, 1, 1

Solution

The sequence 2, 4, 8, 16 has a constant ratio of 2, making it a geometric progression.

Correct Answer: C — 2, 4, 8, 16

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Q. Which of the following statements about a geometric progression is true?

A. The ratio of consecutive terms is constant.
B. The difference between consecutive terms is constant.
C. The sum of the terms is always positive.
D. The first term is always the largest.

Solution

In a geometric progression, the ratio of consecutive terms is indeed constant, which defines the progression.

Correct Answer: A — The ratio of consecutive terms is constant.

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Q. Which of the following statements about geometric progressions is true?

A. The ratio of consecutive terms is constant.
B. The sum of terms is always positive.
C. The first term must be greater than the second.
D. The common ratio can only be an integer.

Solution

In a geometric progression, the ratio of consecutive terms is indeed constant, which defines the progression.

Correct Answer: A — The ratio of consecutive terms is constant.

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Geometric Progression (GP) MCQ & Objective Questions

Geometric Progression (GP) is a crucial topic in mathematics that frequently appears in school and competitive exams. Mastering GP concepts can significantly enhance your problem-solving skills and boost your scores. By practicing MCQs and objective questions, you can solidify your understanding and prepare effectively for your exams. This section is designed to provide you with essential practice questions and important questions that will aid in your exam preparation.

What You Will Practise Here

Definition and basic properties of Geometric Progression (GP)
Formulas related to the nth term and sum of the first n terms
Applications of GP in real-life scenarios
Common types of problems involving GP
Identifying GP from given sequences
Relationship between GP and other mathematical concepts
Diagrams and visual representations of GP

Exam Relevance

Geometric Progression (GP) is a significant topic in various examinations, including CBSE, State Boards, NEET, and JEE. In these exams, you can expect questions that test your understanding of GP concepts, such as finding the nth term, calculating the sum of terms, and solving real-world problems. Common question patterns include multiple-choice questions that require quick calculations and conceptual clarity.

Common Mistakes Students Make

Confusing the formula for the sum of GP with that of Arithmetic Progression (AP)
Misidentifying sequences as GP when they do not have a constant ratio
Overlooking the importance of the first term and common ratio in calculations
Failing to simplify expressions correctly before solving problems

FAQs

Question: What is a Geometric Progression?
Answer: A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Question: How do I find the sum of the first n terms of a GP?
Answer: The sum of the first n terms of a GP can be calculated using the formula: S_n = a(1 - rⁿ) / (1 - r), where 'a' is the first term and 'r' is the common ratio.

Now that you have a clear understanding of Geometric Progression (GP), it's time to put your knowledge to the test! Solve practice MCQs and important questions to enhance your understanding and prepare effectively for your exams. Remember, practice is the key to success!

Geometric Progression (GP)

Geometric Progression (GP) MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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