The second term of an infinte geometric progression is 2 and its sum to infinity is 8. The first term is

Answer

To find the first term of the infinite geometric progression (GP), we will follow these steps:

### Step 1: Understand the properties of the geometric progression
In a geometric progression, the terms can be expressed as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- And so on...

### Step 2: Set up the equations based on the problem
We are given:
1. The second term \( ar = 2 \)
2. The sum to infinity \( S = \frac{a}{1 - r} = 8 \)

### Step 3: Solve for \( r \) from the second term
From the first equation, we can express \( r \) in terms of \( a \):
\[ ar = 2 \implies r = \frac{2}{a} \]

### Step 4: Substitute \( r \) into the sum to infinity formula
Now, substitute \( r \) into the sum to infinity formula:
\[ S = \frac{a}{1 - r} = 8 \]
Substituting \( r = \frac{2}{a} \):
\[ \frac{a}{1 - \frac{2}{a}} = 8 \]

### Step 5: Simplify the equation
To simplify the equation:
\[ \frac{a}{\frac{a - 2}{a}} = 8 \implies \frac{a^2}{a - 2} = 8 \]

### Step 6: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ a^2 = 8(a - 2) \]

### Step 7: Expand and rearrange the equation
Expanding the right side:
\[ a^2 = 8a - 16 \]
Rearranging gives:
\[ a^2 - 8a + 16 = 0 \]

### Step 8: Factor the quadratic equation
This can be factored as:
\[ (a - 4)^2 = 0 \]

### Step 9: Solve for \( a \)
Setting the factor equal to zero gives:
\[ a - 4 = 0 \implies a = 4 \]

### Conclusion
The first term \( a \) of the infinite geometric progression is \( 4 \).

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