Find the sum of 50 terms of the A.P. 1 + 4 + 7 + ....

Explanation

To find the sum of the first 50 terms of the arithmetic progression (A.P.) given by 1, 4, 7, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is 1.
To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 4 - 1 = 3 \]

### Step 2: Determine the number of terms
We are asked to find the sum of the first 50 terms, so \( n = 50 \).

### Step 3: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum \( S_n \) of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Substituting the values we have:
- \( n = 50 \)
- \( a = 1 \)
- \( d = 3 \)

### Step 4: Substitute the values into the formula
\[ S_{50} = \frac{50}{2} \times (2 \times 1 + (50 - 1) \times 3) \]

### Step 5: Simplify the expression
Calculating inside the parentheses first:
\[ 2 \times 1 = 2 \]
\[ 50 - 1 = 49 \]
\[ 49 \times 3 = 147 \]
Now, substituting back:
\[ S_{50} = 25 \times (2 + 147) \]
\[ S_{50} = 25 \times 149 \]

### Step 6: Calculate the final sum
Now, calculate \( 25 \times 149 \):
\[ S_{50} = 3725 \]

### Final Answer
The sum of the first 50 terms of the A.P. is \( \boxed{3725} \).

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