A non constant arithmatic progression has common difference d and first term is `(1- ad)` If the sum of the first 20 term is 20, then the value of a is equal to :

Options

A. `2/19`
B. `19/2`
C. `2/9`
D. `9/2`

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the first term and common difference
The first term of the arithmetic progression (AP) is given as:
\[ a_1 = 1 - ad \]
The common difference is denoted as \( d \).

### Step 2: Write the formula for the sum of the first n terms of an AP
The sum \( S_n \) of the first \( n \) terms of an arithmetic progression can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
For our case, \( n = 20 \), so:
\[ S_{20} = \frac{20}{2} \times (2a_1 + (20 - 1)d) \]
This simplifies to:
\[ S_{20} = 10 \times (2a_1 + 19d) \]

### Step 3: Substitute the known values into the sum formula
We know that the sum of the first 20 terms is 20:
\[ 10 \times (2(1 - ad) + 19d) = 20 \]

### Step 4: Simplify the equation
Dividing both sides by 10 gives:
\[ 2(1 - ad) + 19d = 2 \]
Expanding the left side:
\[ 2 - 2ad + 19d = 2 \]

### Step 5: Rearranging the equation
Subtracting 2 from both sides:
\[ -2ad + 19d = 0 \]

### Step 6: Factor out d
Factoring out \( d \) from the left side:
\[ d(-2a + 19) = 0 \]

### Step 7: Solve for a
Since the progression is non-constant, \( d \neq 0 \). Therefore, we can set the factor equal to zero:
\[ -2a + 19 = 0 \]
Solving for \( a \):
\[ 2a = 19 \]
\[ a = \frac{19}{2} \]

### Final Answer
The value of \( a \) is:
\[ a = \frac{19}{2} \]

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