The sum of first 16 terms of the A.P.: 10,6,2,... Is

Options

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

Answer

To find the sum of the first 16 terms of the arithmetic progression (A.P.) given by 10, 6, 2, ..., 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 \( 10 \).
- The common difference \( d \) can be calculated by subtracting the first term from the second term:
\[ d = 6 - 10 = -4 \]
(You can also verify this by checking the difference between the second and third terms: \( 2 - 6 = -4 \)).

### 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 given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Where:
- \( n \) is the number of terms (in this case, \( n = 16 \)),
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 3: Substitute the values into the formula
Substituting \( n = 16 \), \( a = 10 \), and \( d = -4 \) into the formula:
\[ S_{16} = \frac{16}{2} \times (2 \times 10 + (16 - 1)(-4)) \]

### Step 4: Simplify the expression
Calculating step-by-step:
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 these values back into the equation:
\[ S_{16} = 8 \times (20 - 60) \]
6. Calculate \( 20 - 60 = -40 \).
7. Finally, calculate \( 8 \times (-40) = -320 \).

### Final Answer
The sum of the first 16 terms of the A.P. is:
\[ S_{16} = -320 \]

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