Sum of 100 terms of A.P.:
2,4,6,Is …………………….

Answer

To find the sum of the first 100 terms of the arithmetic progression (A.P.) given as 2, 4, 6, ..., we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a \) is the first number in the sequence, which is \( 2 \).
- The common difference \( d \) is found by subtracting the first term from the second term:
\[ d = 4 - 2 = 2 \]

### Step 2: 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:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Substitute the values into the formula
For this problem, we need to find the sum of the first 100 terms, so \( n = 100 \):
- \( a = 2 \)
- \( d = 2 \)

Substituting these values into the formula:
\[ S_{100} = \frac{100}{2} \times (2 \times 2 + (100 - 1) \times 2) \]

### Step 4: Simplify the expression
Calculating each part:
1. \( \frac{100}{2} = 50 \)
2. \( 2 \times 2 = 4 \)
3. \( 100 - 1 = 99 \)
4. \( 99 \times 2 = 198 \)

Now substituting back:
\[ S_{100} = 50 \times (4 + 198) \]
\[ S_{100} = 50 \times 202 \]

### Step 5: Calculate the final result
Now, calculate \( 50 \times 202 \):
\[ S_{100} = 10100 \]

Thus, the sum of the first 100 terms of the A.P. is **10100**.

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