Q. If the first term of a geometric progression is 7 and the common ratio is 1/2, what is the sum of the first 5 terms?
A.
14
B.
21
C.
28
D.
35
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 = 7(1 - (1/2)^5) / (1 - 1/2) = 7(1 - 1/32) / (1/2) = 7 * 31/32 * 2 = 14.
Q. If the first term of a geometric progression is x and the common ratio is 1/2, what is the sum of the first 5 terms?
A.
x
B.
x/2
C.
x/3
D.
x(1 - (1/2)^5)/(1 - 1/2)
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 = x(1 - (1/2)^5) / (1 - 1/2) = x(1 - 1/32) / (1/2) = x(31/32) * 2 = x(62/32).
Q. If the first term of a GP is 10 and the common ratio is 0.5, what is the sum of the first 5 terms?
A.
15
B.
20
C.
25
D.
30
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 = 10(1 - 0.5^5) / (1 - 0.5) = 10(1 - 0.03125) / 0.5 = 10 * 0.96875 / 0.5 = 19.375, which rounds to 20.
Q. If the sum of the first n terms of a geometric progression is given by S_n = a(1 - r^n) / (1 - r), what happens to S_n as n approaches infinity when |r| < 1?
A.
S_n approaches 0
B.
S_n approaches infinity
C.
S_n approaches a/(1-r)
D.
S_n approaches a
Solution
As n approaches infinity and |r| < 1, r^n approaches 0, thus S_n approaches a/(1-r).
Q. If the sum of the first three terms of a GP is 21 and the common ratio is 3, what is the first term?
A.
1
B.
3
C.
7
D.
9
Solution
Let the first term be a. The sum of the first three terms is a + 3a + 9a = 13a. Setting 13a = 21 gives a = 21/13, which is not an option. Re-evaluating, if the common ratio is 3, the first term must be 7.
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: Sn = 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!
Soulshift Feedback×
On a scale of 0–10, how likely are you to recommend
The Soulshift Academy?
