RD Sharma Class 11 Solutions Chapter 19 - Arithmetic Progressions (Ex 19.4) Exercise 19.4 - Free PDF
FAQs on RD Sharma Class 11 Solutions Chapter 19 - Arithmetic Progressions (Ex 19.4) Exercise 19.4
1. How do you solve questions in RD Sharma Class 11 Exercise 19.4 that ask for the sum of an AP?
To find the sum of an Arithmetic Progression (AP) in this exercise, you typically use one of two key formulas. The primary steps are:
- Identify the first term (a), the common difference (d), and the number of terms (n) from the question.
- If the last term is not known, use the formula: Sn = n/2 [2a + (n-1)d].
- If the last term (l) is known, use the more direct formula: Sn = n/2 (a + l).
- Substitute the values carefully into the chosen formula to find the required sum.
2. What is the step-by-step method to solve word problems in RD Sharma Chapter 19 involving the sum of an AP?
Word problems in Exercise 19.4 require a structured approach. First, you must translate the problem into mathematical terms by identifying the components of the Arithmetic Progression.
- Step 1: Determine the first term (a), common difference (d), and the number of terms (n). For example, for 'the sum of all even numbers from 2 to 100', a=2, d=2, and n=50.
- Step 2: Choose the appropriate sum formula (Sn) based on the information you have.
- Step 3: Calculate the sum by substituting the values.
- Step 4: Write the final answer with the correct units or context as required by the problem.
3. How do you find the number of terms (n) in an AP when it is not directly given in an Exercise 19.4 problem?
Often, you need to find 'n' before you can calculate the sum. To do this, use the formula for the nth term of an AP: an = a + (n-1)d. Here, an represents the last term of the series. By substituting the known values of the first term (a), the last term (an), and the common difference (d), you can solve this linear equation to find the value of 'n'.
4. Why are there two different formulas for the sum of an AP, and how do I choose the right one for questions in Ex 19.4?
Both formulas for the sum of an AP are derived from the same principle but are used in different scenarios for convenience.
- Use Sn = n/2 [2a + (n-1)d] when you know the first term (a), common difference (d), and the number of terms (n). This is the most fundamental formula.
- Use Sn = n/2 (a + l) when you know the first term (a), the last term (l), and the number of terms (n). This is a helpful shortcut, as the last term 'l' is equivalent to a + (n-1)d. Using it saves a calculation step when the last term is explicitly provided in the question.
5. How do I solve a problem that asks for the sum of 'n' terms of an AP when its nth term (an) is given as a linear expression, like an = 5n - 1?
This is a common problem type in RD Sharma. To solve it, you first need to establish the key characteristics of the AP from the given expression.
- Find the first term (a) by substituting n=1 into the expression: a₁ = 5(1) - 1 = 4.
- Find the second term (a₂) by substituting n=2: a₂ = 5(2) - 1 = 9.
- Calculate the common difference (d): d = a₂ - a₁ = 9 - 4 = 5.
- Now that you have 'a' and 'd', you can use the standard sum formula Sn = n/2 [2a + (n-1)d] to find the sum of 'n' terms.
6. In what situation can the sum of an Arithmetic Progression be zero or negative?
The sum of an AP (Sn) can be zero or negative under specific conditions.
- Zero Sum: The sum is zero when the progression contains a balanced set of positive and negative terms that cancel each other out. This usually occurs if the first term (a) is positive and the common difference (d) is negative.
- Negative Sum: The sum becomes negative when the cumulative value of the negative terms in the series outweighs the cumulative value of the positive terms. This can happen in an AP with a negative common difference or if the entire AP consists of negative numbers.
