The number of terms of an A.P. is odd. The sum of the odd terms `(1 ^(st), 3 ^(rd) etc,)` is 248 and the sum of the even terms is 217. The last term exceeds the first by 56 then :

Options

A. the number of terms is 17
B. the first term is 3
C. the number of terms is 13
D. the first term is 1

Answer

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.) and the given conditions.

### Step 1: Define Variables
Let:
- \( a \) = first term of the A.P.
- \( d \) = common difference
- \( n \) = number of odd terms in the A.P.
- The total number of terms in the A.P. = \( 2n + 1 \) (since the number of terms is odd).

### Step 2: Sum of Odd Terms
The sum of the odd terms (1st, 3rd, 5th, ..., (2n+1)th terms) can be expressed as:
\[ S_{\text{odd}} = \frac{n + 1}{2} \left( 2a + (n - 1) \cdot 2d \right) = 248 \]
This simplifies to:
\[ (n + 1)(a + nd) = 248 \quad \text{(Equation 1)} \]

### Step 3: Sum of Even Terms
The sum of the even terms (2nd, 4th, 6th, ..., 2nth terms) can be expressed as:
\[ S_{\text{even}} = \frac{n}{2} \left( 2(a + d) + (n - 1) \cdot 2d \right) = 217 \]
This simplifies to:
\[ n(a + nd + d) = 217 \quad \text{(Equation 2)} \]

### Step 4: Last Term Condition
The last term of the A.P. exceeds the first term by 56:
\[ (2n + 1)a + 2nd - a = 56 \]
This simplifies to:
\[ 2nd + a = 56 \quad \text{(Equation 3)} \]

### Step 5: Solve the Equations
From Equation 3, we can express \( a \):
\[ a = 56 - 2nd \]

Substituting \( a \) into Equation 1:
\[ (n + 1)(56 - 2nd + nd) = 248 \]
This simplifies to:
\[ (n + 1)(56 - nd) = 248 \]
Expanding gives:
\[ 56n + 56 - n^2d - nd = 248 \]
Rearranging:
\[ -n^2d - nd + 56n - 192 = 0 \quad \text{(Equation 4)} \]

Now substituting \( a \) into Equation 2:
\[ n(56 - 2nd + nd + d) = 217 \]
This simplifies to:
\[ n(56 - nd) = 217 \]
Expanding gives:
\[ 56n - n^2d = 217 \]
Rearranging:
\[ -n^2d + 56n - 217 = 0 \quad \text{(Equation 5)} \]

### Step 6: Solve Equations 4 and 5
Now we have two equations (4 and 5) in terms of \( n \) and \( d \). We can solve these simultaneously.

From Equation 5:
\[ -n^2d = 217 - 56n \]
Substituting into Equation 4:
\[ (217 - 56n) + 56n - 192 = 0 \]
This simplifies to:
\[ 217 - 192 = 0 \]
Thus:
\[ n^2d = 25 \]

### Step 7: Find Values of \( n \) and \( d \)
Using the values derived, we can find \( n \) and \( d \). Let's assume \( n = 7 \) (as it’s an odd number) and substitute back to find \( d \):
\[ 7^2d = 25 \implies d = \frac{25}{49} \]

### Step 8: Find \( a \)
Now substituting \( n \) and \( d \) back into the equation for \( a \):
\[ a = 56 - 2(7)d = 56 - 14 \cdot \frac{25}{49} \]
Calculating gives:
\[ a = 56 - \frac{350}{49} = 56 - 7.14 \approx 48.86 \]

### Conclusion
Thus, the values of \( n \), \( a \), and \( d \) can be verified against the conditions provided in the problem statement.