In ap An=394,A1=3 ,18<n<30 Find S(n-3) and A(n-3)

Given the arithmetic progression (AP) with \(A_n = 394\), \(A_1 = 3\), and \(18 < n < 30\), we need to find \(S_{n-3}\) and \(A_{n-3}\).

First, let's recall the formulas for an arithmetic progression (AP):

1. **General term of an AP:**
   \[
   A_n = A_1 + (n-1)d
   \]
   where \(A_n\) is the nth term, \(A_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

2. **Sum of the first \(m\) terms of an AP:**
   \[
   S_m = \frac{m}{2} (2A_1 + (m-1)d)
   \]
   where \(S_m\) is the sum of the first \(m\) terms.

### Step-by-Step Solution:

1. **Find the common difference \(d\):**

   Given \(A_n = 394\) and \(A_1 = 3\), we can set up the equation:
   \[
   394 = 3 + (n-1)d
   \]
   Solving for \(d\):
   \[
   391 = (n-1)d \implies d = \frac{391}{n-1}
   \]

2. **Find \(n\):**

   Since \(18 < n < 30\), we need \(d\) to be an integer. Let's test values of \(n\) within the range:
   
   For \(n = 29\):
   \[
   d = \frac{391}{29-1} = \frac{391}{28} \approx 13.9643 \quad (\text{not an integer})
   \]

   For \(n = 28\):
   \[
   d = \frac{391}{28-1} = \frac{391}{27} \approx 14.4815 \quad (\text{not an integer})
   \]

   For \(n = 27\):
   \[
   d = \frac{391}{27-1} = \frac{391}{26} \approx 15.0385 \quad (\text{not an integer})
   \]

   For \(n = 26\):
   \[
   d = \frac{391}{26-1} = \frac{391}{25} \approx 15.64 \quad (\text{not an integer})
   \]

   For \(n = 25\):
   \[
   d = \frac{391}{25-1} = \frac{391}{24} \approx 16.2917 \quad (\text{not an integer})
   \]

   For \(n = 24\):
   \[
   d = \frac{391}{24-1} = \frac{391}{23} \approx 17 \quad (\text{integer})
   \]

   Therefore, \(n = 24\) and \(d = 17\).

3. **Find \(A_{n-3}\):**

   \[
   A_{21} = A_1 + (21-1)d = 3 + 20 \cdot 17 = 3 + 340 = 343
   \]

4. **Find \(S_{n-3}\):**

   \[
   S_{21} = \frac{21}{2} (2 \cdot 3 + 20 \cdot 17) = \frac{21}{2} (6 + 340) = \frac{21}{2} \cdot 346 = 21 \cdot 173 = 3633
   \]

### Results:

- \(A_{n-3} = A_{21} = 343\)
- \(S_{n-3} = S_{21} = 3633\)

Would you like more details or have any questions?

Here are 5 related questions you might want to ask:
1. How do you derive the formula for the sum of an arithmetic series?
2. Can you provide an example with a different set of values for \(A_1\) and \(A_n\)?
3. How do you determine if \(d\) is an integer for different values of \(n\)?
4. What are the applications of arithmetic progressions in real life?
5. Can you explain how to find the \(n\)th term in a geometric progression?

**Tip:** When solving for the common difference in an arithmetic progression, always ensure to check if it results in an integer for the given range of \(n\).

Learn how to find S(n-3) and A(n-3) in an arithmetic progression (AP) given A_n = 394, A_1 = 3, and 18 < n < 30. Includes step-by-step solutions using formulas for AP and sum of series.

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