Find the sum of the following A. P. s:
`0.6 , 1.7, 2.8 ,…,` to 100 terms.

Answer

To find the sum of the arithmetic progression (A.P.) given by the terms 0.6, 1.7, 2.8, ..., up to 100 terms, we will follow these steps:

### Step 1: Identify the first term (a) and common difference (d)
- The first term \( a = 0.6 \)
- To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 1.7 - 0.6 = 1.1 \]

### Step 2: Use the formula for the sum of the first n terms of an A.P.
The formula for the sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( S_n \) = sum of the first \( n \) terms
- \( n \) = number of terms
- \( a \) = first term
- \( d \) = common difference

### Step 3: Substitute the values into the formula
Here, we need to find the sum of the first 100 terms, so \( n = 100 \):
\[ S_{100} = \frac{100}{2} \times (2 \times 0.6 + (100 - 1) \times 1.1) \]

### Step 4: Simplify the expression
Calculating the components:
- \( 2a = 2 \times 0.6 = 1.2 \)
- \( n - 1 = 100 - 1 = 99 \)
- \( (n - 1)d = 99 \times 1.1 = 108.9 \)

Now substitute these values back into the equation:
\[ S_{100} = 50 \times (1.2 + 108.9) \]
\[ S_{100} = 50 \times 110.1 \]

### Step 5: Calculate the final sum
Now, calculate \( 50 \times 110.1 \):
\[ S_{100} = 5505 \]

### Final Answer:
The sum of the first 100 terms of the given A.P. is **5505**.

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