What is the sum of first 10 terms of the A.P. 15, 10 ,5 ….?

Options

A. `-75`
B. `-125`
C. 75
D. 125

Answer

To find the sum of the first 10 terms of the arithmetic progression (A.P.) given as 15, 10, 5, ..., 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:
\[ A = 15 \]

To find the common difference \( D \), we subtract the first term from the second term:
\[ D = 10 - 15 = -5 \]

### 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 \( S_n \) of an A.P. is:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]

### Step 3: Substitute the values into the formula
We want to find the sum of the first 10 terms, so \( n = 10 \):
\[ S_{10} = \frac{10}{2} \times (2 \times 15 + (10 - 1)(-5)) \]

### Step 4: Simplify the expression
Calculating \( S_{10} \):
1. Calculate \( \frac{10}{2} = 5 \)
2. Calculate \( 2 \times 15 = 30 \)
3. Calculate \( (10 - 1) = 9 \)
4. Calculate \( 9 \times (-5) = -45 \)

Now substitute back into the equation:
\[ S_{10} = 5 \times (30 - 45) \]
\[ S_{10} = 5 \times (-15) \]

### Step 5: Final calculation
\[ S_{10} = -75 \]

Thus, the sum of the first 10 terms of the A.P. is:
\[ \boxed{-75} \]

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