The sum of first 16 terms of the AP 10, 6, 2, … is

Options

A. -320
B. 320
C. -352
D. -400

Explanation

To find the sum of the first 16 terms of the arithmetic progression (AP) given as 10, 6, 2, ..., we can follow these steps:

### Step 1: Identify the first term and common difference
- The first term \( a \) is the first number in the sequence, which is \( 10 \).
- The common difference \( d \) can be calculated as the difference between the second term and the first term:
\[ d = a_2 - a_1 = 6 - 10 = -4 \]

### Step 2: Use the formula for the sum of the first \( n \) terms of an AP
The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression is:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Where:
- \( n \) is the number of terms,
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Substitute the values into the formula
We need to find the sum of the first 16 terms, so we set \( n = 16 \):
\[ S_{16} = \frac{16}{2} \times (2 \times 10 + (16 - 1)(-4)) \]

### Step 4: Simplify the expression
Calculating \( S_{16} \):
1. Calculate \( \frac{16}{2} = 8 \).
2. Calculate \( 2 \times 10 = 20 \).
3. Calculate \( 16 - 1 = 15 \).
4. Calculate \( 15 \times (-4) = -60 \).
5. Now substitute back into the equation:
\[ S_{16} = 8 \times (20 - 60) \]
6. Simplify \( 20 - 60 = -40 \).
7. Finally calculate \( S_{16} = 8 \times (-40) = -320 \).

### Final Answer
The sum of the first 16 terms of the AP is \( -320 \).

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