if second terms of a GP is 2 and the sun of its infinite terms is , then its first term is

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 2
D. 4

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The second term of the geometric progression (GP) is 2.
- The sum of the infinite terms of the GP is 8.

### Step 2: Use the formula for the second term
In a GP, the second term can be expressed as:
\[ \text{Second term} = ar = 2 \]
where \( a \) is the first term and \( r \) is the common ratio.

### Step 3: Express \( a \) in terms of \( r \)
From the equation \( ar = 2 \), we can express \( a \) as:
\[ a = \frac{2}{r} \]

### Step 4: Use the formula for the sum of infinite terms
The sum of an infinite GP is given by the formula:
\[ S = \frac{a}{1 - r} \]
We know that the sum is 8, so we can write:
\[ \frac{a}{1 - r} = 8 \]

### Step 5: Substitute \( a \) in the sum formula
Substituting \( a = \frac{2}{r} \) into the sum formula:
\[ \frac{\frac{2}{r}}{1 - r} = 8 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 2 = 8r(1 - r) \]
Expanding this:
\[ 2 = 8r - 8r^2 \]

### Step 7: Rearrange the equation
Rearranging the equation leads to:
\[ 8r^2 - 8r + 2 = 0 \]

### Step 8: Simplify the equation
Dividing the entire equation by 2:
\[ 4r^2 - 4r + 1 = 0 \]

### Step 9: Factor the quadratic equation
This can be factored as:
\[ (2r - 1)^2 = 0 \]

### Step 10: Solve for \( r \)
Setting the factor equal to zero gives:
\[ 2r - 1 = 0 \]
Thus,
\[ r = \frac{1}{2} \]

### Step 11: Substitute \( r \) back to find \( a \)
Now, substitute \( r \) back into the equation \( a = \frac{2}{r} \):
\[ a = \frac{2}{\frac{1}{2}} = 4 \]

### Conclusion
The first term \( a \) of the GP is:
\[ \boxed{4} \]