Find the sum of:
10 terms of 5 +8 + 11+ ....,

Answer

To find the sum of the first 10 terms of the arithmetic progression (AP) given by the sequence 5, 8, 11, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
- The first term \( a \) of the AP is 5.
- The common difference \( d \) can be calculated as follows:
\[ d = 8 - 5 = 3 \]

### 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 AP
The formula for the sum \( S_n \) of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 4: Substitute the values into the formula
Substituting \( n = 10 \), \( a = 5 \), and \( d = 3 \) into the formula:
\[ S_{10} = \frac{10}{2} \times (2 \times 5 + (10 - 1) \times 3) \]

### Step 5: Simplify the expression
Calculating inside the parentheses:
\[ S_{10} = 5 \times (10 + 9 \times 3) \]
Calculating \( 9 \times 3 \):
\[ 9 \times 3 = 27 \]
So,
\[ S_{10} = 5 \times (10 + 27) = 5 \times 37 \]

### Step 6: Final calculation
Now, calculate \( 5 \times 37 \):
\[ S_{10} = 185 \]

### Final Answer
The sum of the first 10 terms of the sequence is \( 185 \).

---