Find the sum of 23 terms of the AP 17,12,7,2,-3,….

Answer

To find the sum of the first 23 terms of the arithmetic progression (AP) given by the sequence 17, 12, 7, 2, -3, ..., 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 \( 17 \).
- To find the common difference \( D \), we subtract the first term from the second term:
\[ D = 12 - 17 = -5 \]

### Step 2: Identify the number of terms (n)
- We need to find the sum of the first \( n = 23 \) terms.

### Step 3: Use the formula for the sum of the first n terms of an AP
The formula to calculate the sum \( S_n \) of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
Substituting the values we have:
- \( n = 23 \)
- \( A = 17 \)
- \( D = -5 \)

### Step 4: Substitute the values into the formula
\[ S_{23} = \frac{23}{2} \times (2 \times 17 + (23 - 1)(-5)) \]

### Step 5: Calculate the expression inside the parentheses
First, calculate \( 2A \):
\[ 2A = 2 \times 17 = 34 \]
Next, calculate \( (n - 1)D \):
\[ (n - 1)D = 22 \times (-5) = -110 \]
Now, combine these results:
\[ 2A + (n - 1)D = 34 - 110 = -76 \]

### Step 6: Substitute back into the sum formula
\[ S_{23} = \frac{23}{2} \times (-76) \]

### Step 7: Calculate the sum
\[ S_{23} = \frac{23 \times -76}{2} = \frac{-1748}{2} = -874 \]

### Final Answer
The sum of the first 23 terms of the AP is \( -874 \).

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