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 will use the formulas related to Arithmetic Progressions (A.P.).

### Given:
- First term \( a = 5 \)
- Common difference \( d = 3 \)
- \( a_n = 50 \) (the nth term)

### Step 1: Find \( n \) using the nth term formula
The formula for the nth term 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
First, subtract 5 from both sides:
\[ 50 - 5 = (n-1) \cdot 3 \]
\[ 45 = (n-1) \cdot 3 \]

### Step 3: Divide by the common difference
Now, divide both sides by 3:
\[ \frac{45}{3} = n - 1 \]
\[ 15 = n - 1 \]

### Step 4: Solve for \( n \)
Add 1 to both sides to find \( n \):
\[ n = 15 + 1 \]
\[ n = 16 \]

### Step 5: Find the sum of the first \( n \) terms \( S_n \)
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \cdot (2a + (n-1) \cdot d) \]
Substituting \( n = 16 \), \( a = 5 \), and \( d = 3 \):
\[ S_{16} = \frac{16}{2} \cdot (2 \cdot 5 + (16 - 1) \cdot 3) \]

### Step 6: Simplify the sum formula
Calculate:
\[ S_{16} = 8 \cdot (10 + 15 \cdot 3) \]
\[ S_{16} = 8 \cdot (10 + 45) \]
\[ S_{16} = 8 \cdot 55 \]

### Step 7: Calculate \( S_{16} \)
Finally, calculate:
\[ S_{16} = 440 \]

### Final Answers:
- \( n = 16 \)
- \( S_{16} = 440 \)

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