In an A. P. :
given `a = 5, d = 3, a _(n) = 50,` find n and `S _(n).`

Answer

To solve the problem step by step, we need to find the value of \( n \) and \( S_n \) in the given Arithmetic Progression (A.P.) where \( a = 5 \), \( d = 3 \), and \( a_n = 50 \).

### Step 1: Use the formula for the \( n \)-th term of an A.P.
The formula for the \( n \)-th term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 50 = 5 + (n - 1) \cdot 3 \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 50 = 5 + 3(n - 1) \]
Subtract 5 from both sides:
\[ 50 - 5 = 3(n - 1) \]
\[ 45 = 3(n - 1) \]

### Step 3: Solve for \( n \)
Next, divide both sides by 3:
\[ 15 = n - 1 \]
Now, add 1 to both sides:
\[ n = 15 + 1 = 16 \]

### Step 4: Find \( S_n \) using the sum formula
Now that we have \( n = 16 \), we can find \( S_n \) using the formula for the sum of the first \( n \) terms of an A.P.:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1) \cdot d) \]
Substituting the values:
\[ S_{16} = \frac{16}{2} \cdot (2 \cdot 5 + (16 - 1) \cdot 3) \]

### Step 5: Calculate \( S_n \)
Now, calculate the terms:
\[ S_{16} = 8 \cdot (10 + 15 \cdot 3) \]
Calculate \( 15 \cdot 3 \):
\[ 15 \cdot 3 = 45 \]
Now substitute back:
\[ S_{16} = 8 \cdot (10 + 45) = 8 \cdot 55 \]
Finally, calculate \( 8 \cdot 55 \):
\[ S_{16} = 440 \]

### Final Answer
Thus, the values are:
- \( n = 16 \)
- \( S_n = 440 \)

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