RD Sharma Class 11 Solutions Chapter 20 By Vedantu
FAQs on RD Sharma Class 11 Solutions Chapter 20 - Geometric Progressions (Ex 20.1) Exercise 20.1
1. What are the key concepts required to solve problems in RD Sharma Class 11 Solutions for Chapter 20, Exercise 20.1?
Exercise 20.1 primarily focuses on the fundamentals of a Geometric Progression (GP). To solve the problems, you must understand:
- The definition of a GP: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).
- How to find the common ratio: By dividing any term by its preceding term (r = a_n / a_n-1).
- The formula for the nth term of a GP: a_n = ar^(n-1), where 'a' is the first term and 'r' is the common ratio.
This exercise builds the foundation for more advanced GP concepts.
2. How do you find the 12th term of a GP if the sequence is 2, 4, 8, ... using the method from Chapter 20?
To solve this typical Exercise 20.1 problem, you need to follow these steps as per the CBSE syllabus methodology:
- Identify the first term (a): In the sequence 2, 4, 8, ..., the first term is a = 2.
- Calculate the common ratio (r): Divide the second term by the first term: r = 4 / 2 = 2. You can verify this with the next pair: 8 / 4 = 2. So, r = 2.
- Identify the term to find (n): We need to find the 12th term, so n = 12.
- Apply the nth term formula: Use the formula a_n = ar^(n-1). Substitute the values: a_12 = 2 × (2)^(12-1) = 2 × 2^11 = 2^12.
- Calculate the final value: 2^12 = 4096. Thus, the 12th term is 4096.
3. What is the correct step-by-step method to verify if a given sequence is a Geometric Progression, as required in Exercise 20.1?
To correctly verify if a sequence is a GP, you must check for a constant common ratio across the sequence. The steps are:
- Calculate the ratio of the second term to the first term (a₂/a₁).
- Calculate the ratio of the third term to the second term (a₃/a₂).
- Continue this for at least one more pair if available (e.g., a₄/a₃).
- If all these calculated ratios are exactly the same, then the sequence is a GP. If any ratio is different, it is not a GP. Simply checking one pair is not sufficient proof.
4. Can a Geometric Progression have a negative common ratio? How does it affect the terms of the sequence?
Yes, a Geometric Progression can absolutely have a negative common ratio (r < 0). This is a key concept tested in Chapter 20. When 'r' is negative, the terms of the GP will alternate in sign. For example, if the first term 'a' is 3 and the common ratio 'r' is -2, the sequence will be 3, -6, 12, -24, and so on. This alternating pattern is a distinct characteristic of a GP with a negative common ratio.
5. What is the most common mistake students make when solving for an unknown term in RD Sharma's Exercise 20.1?
A frequent mistake is incorrectly applying the exponent in the nth term formula, a_n = ar^(n-1). Students often calculate (ar)^(n-1) instead of the correct a × (r^(n-1)). Remember, the exponent (n-1) applies only to the common ratio 'r', not to the first term 'a'. Always calculate the power of 'r' first, and then multiply the result by 'a' to get the correct answer.
6. How do you find the first term and common ratio if two non-consecutive terms of a GP are given (e.g., the 4th term and the 8th term)?
This is a classic problem type in Chapter 20. The correct method involves setting up a system of two equations using the nth term formula:
- Let the given terms be a_m = x and a_n = y.
- Write the two equations using the formula a_n = ar^(n-1): x = ar^(m-1) and y = ar^(n-1).
- Divide the second equation by the first: (y/x) = (ar^(n-1)) / (ar^(m-1)).
- This simplifies to y/x = r^(n-m). From this equation, you can solve for the common ratio 'r'.
- Once 'r' is found, substitute its value back into either of the original equations (x = ar^(m-1) or y = ar^(n-1)) to solve for the first term 'a'.
7. Does Exercise 20.1 of RD Sharma Class 11 Maths cover problems on the sum of a GP?
No, Exercise 20.1 strictly focuses on the definition and the nth term of a Geometric Progression. Questions in this exercise involve identifying a GP, finding its common ratio, and calculating a specific term (like the 10th term or 20th term). The concepts of the sum of n terms of a GP, the sum of an infinite GP, and Geometric Mean are introduced in subsequent exercises of Chapter 20.
