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 that \( a_1, a_2, a_3, \ldots, a_n \) are in Arithmetic Progression (AP), we know that:
- \( a_1 = 1 \)
- \( a_n = 300 \)
- The common difference \( d \) is an integer.

The formula for the \( n \)-th term of an AP is given by:
\[ a_n = a_1 + (n - 1) \cdot d \]

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

### Step 3: Factor 299
Next, we need to factor 299. The prime factorization of 299 is:
\[ 299 = 13 \times 23 \]
Thus, the pairs of factors of 299 are:
- \( (1, 299) \)
- \( (13, 23) \)
- \( (23, 13) \)
- \( (299, 1) \)

### Step 4: Determine possible values for \( n \) and \( d \)
From the equation \( 299 = (n - 1) \cdot d \), we can set:
1. \( n - 1 = 1 \) and \( d = 299 \) → \( n = 2 \) (not in range)
2. \( n - 1 = 13 \) and \( d = 23 \) → \( n = 14 \) (not in range)
3. \( n - 1 = 23 \) and \( d = 13 \) → \( n = 24 \) (valid)
4. \( n - 1 = 299 \) and \( d = 1 \) → \( n = 300 \) (not in range)

The only valid pair that fits the range \( n \in [15, 50] \) is:
\[ n = 24, \quad d = 13 \]

### Step 5: Find \( a_{n-4} \)
Now, we need to find \( a_{n-4} \):
\[ n - 4 = 24 - 4 = 20 \]
Using the formula for the \( n \)-th term:
\[ a_{20} = a_1 + (20 - 1) \cdot d = 1 + 19 \cdot 13 \]
Calculating this gives:
\[ a_{20} = 1 + 247 = 248 \]

### Step 6: Find \( S_{n-4} \)
Next, we calculate \( S_{n-4} \) (the sum of the first \( n-4 \) terms):
\[ S_{20} = \frac{n}{2} \cdot (a_1 + a_{20}) = \frac{20}{2} \cdot (1 + 248) \]
Calculating this gives:
\[ S_{20} = 10 \cdot 249 = 2490 \]

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