Find the sum of the following A. P. s:
`2,7,12,…,` to 10 terms.

Answer

To find the sum of the given arithmetic progression (A.P.) \(2, 7, 12, \ldots\) up to 10 terms, we will follow these steps:

### Step 1: Identify the first term and the common difference
- The first term \(a\) is \(2\).
- To find the common difference \(d\), subtract the first term from the second term:
\[ d = 7 - 2 = 5 \]

### Step 2: Determine the number of terms
- We need to find the sum of the first \(n = 10\) terms.

### 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:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 4: Substitute the values into the formula
Substituting \(n = 10\), \(a = 2\), and \(d = 5\) into the formula:
\[ S_{10} = \frac{10}{2} \times (2 \times 2 + (10 - 1) \times 5) \]

### Step 5: Simplify the expression
Calculating step-by-step:
1. Calculate \(2a\):
\[ 2a = 2 \times 2 = 4 \]
2. Calculate \((n - 1)d\):
\[ (10 - 1) \times 5 = 9 \times 5 = 45 \]
3. Now substitute back into the sum formula:
\[ S_{10} = 5 \times (4 + 45) \]
4. Simplify inside the parentheses:
\[ 4 + 45 = 49 \]
5. Finally, calculate \(S_{10}\):
\[ S_{10} = 5 \times 49 = 245 \]

### Final Answer
The sum of the first 10 terms of the A.P. is \(245\).

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