RD Sharma Class 10 Solutions Chapter 9 - Arithmetic Progressions (Ex 9.3) Exercise 9.3 - Free PDF
FAQs on RD Sharma Class 10 Solutions Chapter 9 - Exercise 9.3
1. What are the key formulas from Chapter 9 required to solve the problems in RD Sharma Class 10 Exercise 9.3?
To solve the problems in Exercise 9.3, which primarily deals with the sum of terms in an Arithmetic Progression, you will mainly use the following two formulas:
The formula for the sum of the first 'n' terms: S_n = n/2[2a + (n-1)d], where 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.
An alternative formula for the sum when the last term ('l') is known: S_n = n/2(a + l).
You may also need the formula for the nth term, a_n = a + (n-1)d, to find the number of terms 'n' or the last term 'l' before you can calculate the sum.
2. What is the correct step-by-step method to find the sum of an AP as per the solutions for Exercise 9.3?
The correct step-by-step method to find the sum of an Arithmetic Progression (AP) is as follows:
Step 1: Identify the known values from the problem, which are typically the first term (a), the common difference (d), and the number of terms (n).
Step 2: Choose the appropriate sum formula. If the last term (l) is known, use S_n = n/2(a + l). Otherwise, use S_n = n/2[2a + (n-1)d].
Step 3: Substitute the identified values of a, d, and n into the chosen formula.
Step 4: Perform the calculations carefully, paying close attention to the order of operations and signs, to find the final sum (S_n).
3. How do I decide whether to use the formula S_n = n/2(a + l) or S_n = n/2(2a + (n-1)d) for a given problem?
Your choice of formula depends on the information provided in the question. You should use:
S_n = n/2(a + l) when you know the first term (a), the last term (l), and the total number of terms (n). This is the more direct and quicker method in such cases.
S_n = n/2[2a + (n-1)d] when you know the first term (a), the common difference (d), and the number of terms (n), but the last term is not given.
4. How are the nth term (a_n) and the sum of n terms (S_n) of an AP related? Can I find a specific term using only the sum formulas?
Yes, the nth term and the sum of n terms are closely related. The nth term of an AP is the difference between the sum of the first 'n' terms and the sum of the first 'n-1' terms. This gives a very useful relationship: a_n = S_n - S_{n-1}. This means if you know the general formula for the sum of 'n' terms of an AP, you can find any specific term without knowing the first term or common difference directly.
5. What are the common mistakes to avoid when solving problems from Exercise 9.3 involving the sum of an AP?
When solving for the sum of an AP, students often make a few common errors. Be careful to avoid:
Calculation Errors: Simple mistakes in arithmetic, especially with negative signs for the common difference (d).
Formula Errors: Forgetting to multiply by 'n/2' or using 'n' instead of '(n-1)' inside the bracket of the formula S_n = n/2[2a + (n-1)d].
Incorrect 'n': Miscalculating the number of terms 'n' before applying the sum formula. Always double-check how you found 'n'.
Quadratic Equation Mistakes: When finding 'n' from a given sum, you may get a quadratic equation. Ensure you solve it correctly and choose the valid positive integer value for 'n'.
6. How do I solve a problem where the sum (S_n) is given, and I need to find the number of terms (n)?
To find the number of terms (n) when the sum (S_n) is given, you should follow these steps:
1. Write down the formula S_n = n/2[2a + (n-1)d].
2. Substitute the known values of S_n, the first term (a), and the common difference (d) into the equation.
3. Simplify the equation. This will typically result in a quadratic equation of the form An² + Bn + C = 0.
4. Solve this quadratic equation for 'n' using methods like factorization or the quadratic formula.
5. Since 'n' represents the number of terms, it must be a positive integer. Discard any negative or fractional solutions.
7. In some problems, the sum of an AP can be zero or negative. What does this practically mean?
A zero or negative sum in an AP has a practical meaning. A zero sum (S_n = 0) implies that the sum of all the positive terms in the progression is perfectly cancelled out by the sum of all the negative terms. A negative sum (S_n < 0) indicates that the cumulative value of the negative terms in the sequence is greater than the cumulative value of the positive terms. This often happens in progressions that start with positive terms but have a negative common difference.
