In an A.P:
given a= 7 ,`a_(13)=35` , find d and `S_(13)`

Answer

To solve the problem step by step, we will find the common difference \( d \) and the sum of the first 13 terms \( S_{13} \) of the arithmetic progression (A.P.) given that the first term \( a = 7 \) and the 13th term \( a_{13} = 35 \).

### Step 1: Use the formula for the nth term of an A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1)d \]
Here, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

### Step 2: Substitute the known values into the formula
For the 13th term:
\[ a_{13} = a + (13 - 1)d \]
Substituting the known values:
\[ 35 = 7 + (12)d \]

### Step 3: Simplify the equation
Now, we can rearrange the equation to solve for \( d \):
\[ 35 - 7 = 12d \]
\[ 28 = 12d \]

### Step 4: Solve for \( d \)
Now, divide both sides by 12:
\[ d = \frac{28}{12} \]
Simplifying this fraction:
\[ d = \frac{7}{3} \]

### Step 5: Calculate the sum of the first 13 terms \( S_{13} \)
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \left(2a + (n - 1)d\right) \]
Substituting \( n = 13 \), \( a = 7 \), and \( d = \frac{7}{3} \):
\[ S_{13} = \frac{13}{2} \left(2 \cdot 7 + (13 - 1) \cdot \frac{7}{3}\right) \]

### Step 6: Simplify the expression inside the parentheses
Calculating \( 2 \cdot 7 \):
\[ 2 \cdot 7 = 14 \]
Calculating \( (13 - 1) \cdot \frac{7}{3} \):
\[ 12 \cdot \frac{7}{3} = \frac{84}{3} = 28 \]
Now, substituting back into the sum formula:
\[ S_{13} = \frac{13}{2} \left(14 + 28\right) \]
\[ S_{13} = \frac{13}{2} \cdot 42 \]

### Step 7: Final calculation of \( S_{13} \)
Calculating \( \frac{13 \cdot 42}{2} \):
\[ S_{13} = \frac{546}{2} = 273 \]

### Final Answers
- The common difference \( d = \frac{7}{3} \)
- The sum of the first 13 terms \( S_{13} = 273 \)

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