If `a_1,a_2,a_3,…….a_n` are in Arithmetic Progression, whose common difference is an integer such that`a_1=1,a_n=300` and `n in[15,50]` then `(S_(n-4),a_(n-4))` is

Options

A. `(2491,247)`
B. `(2490,248)`
C. `(2590,249)`
D. `(248,2490)`

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Understand the Arithmetic Progression (AP)
Given:
- \( a_1 = 1 \)
- \( a_n = 300 \)
- \( n \) is in the range [15, 50]

In an arithmetic progression, the \( n \)-th term can be expressed as:
\[ a_n = a_1 + (n - 1) \cdot d \]
where \( d \) is the common difference.

### Step 2: Set up the equation using the given values
Substituting the known values into the formula:
\[ 300 = 1 + (n - 1) \cdot d \]
This simplifies to:
\[ (n - 1) \cdot d = 299 \]

### Step 3: Factor 299
Next, we need to find the integer factors of 299. The prime factorization of 299 is:
\[ 299 = 13 \cdot 23 \]
Thus, the pairs of factors (considering \( n - 1 \) and \( d \)) can be:
1. \( n - 1 = 1 \) and \( d = 299 \)
2. \( n - 1 = 13 \) and \( d = 23 \)
3. \( n - 1 = 23 \) and \( d = 13 \)
4. \( n - 1 = 299 \) and \( d = 1 \)

### Step 4: Determine valid \( n \) values
We need \( n \) to be in the range [15, 50]. Let's calculate \( n \) for each factor pair:
1. \( n - 1 = 1 \) → \( n = 2 \) (not valid)
2. \( n - 1 = 13 \) → \( n = 14 \) (not valid)
3. \( n - 1 = 23 \) → \( n = 24 \) (valid)
4. \( n - 1 = 299 \) → \( n = 300 \) (not valid)

Thus, the only valid option is:
\[ n = 24 \quad \text{and} \quad d = 13 \]

### Step 5: Calculate \( a_{n-4} \)
Now we need to find \( a_{n-4} \), which is \( a_{20} \):
\[ a_{20} = a_1 + (20 - 1) \cdot d \]
Substituting the values:
\[ a_{20} = 1 + 19 \cdot 13 = 1 + 247 = 248 \]

### Step 6: Calculate \( S_{n-4} \)
Next, we calculate \( S_{n-4} = S_{20} \):
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting \( n = 20 \), \( a = 1 \), and \( d = 13 \):
\[ S_{20} = \frac{20}{2} \cdot (2 \cdot 1 + (20 - 1) \cdot 13) \]
This simplifies to:
\[ S_{20} = 10 \cdot (2 + 19 \cdot 13) = 10 \cdot (2 + 247) = 10 \cdot 249 = 2490 \]

### Final Result
Thus, the values of \( (S_{n-4}, a_{n-4}) \) are:
\[ (S_{20}, a_{20}) = (2490, 248) \]