Find the sum of indicated number of terms of the following arithmetic progression: 16,11,6,………….23 terms.

Answer

To find the sum of the first 23 terms of the arithmetic progression (AP) given by the sequence 16, 11, 6, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
- The first term \( a \) is 16.
- The common difference \( d \) can be calculated as follows:
\[ d = 11 - 16 = -5 \]

### Step 2: Determine the number of terms
- The number of terms \( n \) is given as 23.

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

### Step 4: Substitute the values into the formula
Substituting \( n = 23 \), \( a = 16 \), and \( d = -5 \):
\[ S_{23} = \frac{23}{2} \times (2 \times 16 + (23 - 1) \times (-5)) \]

### Step 5: Simplify the expression
Calculating inside the parentheses:
1. Calculate \( 2a \):
\[ 2 \times 16 = 32 \]
2. Calculate \( (n - 1) \times d \):
\[ (23 - 1) \times (-5) = 22 \times (-5) = -110 \]
3. Combine these results:
\[ 32 - 110 = -78 \]

### Step 6: Final calculation of \( S_{23} \)
Now substitute back into the sum formula:
\[ S_{23} = \frac{23}{2} \times (-78) \]
Calculating this:
1. Multiply:
\[ S_{23} = \frac{23 \times -78}{2} \]
\[ = \frac{-1794}{2} = -897 \]

### Conclusion
The sum of the first 23 terms of the given arithmetic progression is:
\[ \boxed{-897} \]