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, we can follow these steps:

### Step 1: Define the terms of the geometric progression
Let the first term be \( a \) and the common ratio be \( r \). The terms of the geometric progression can be represented as:
- First term: \( a \)
- Second term: \( ar \)
- Third term: \( ar^2 \)
- ...

### Step 2: Use the information given
From the problem, we know:
- The second term \( ar = 2 \)
- The sum to infinity \( S_\infty = \frac{a}{1 - r} = 8 \)

### Step 3: Express \( r \) in terms of \( a \)
From the second term equation:
\[ ar = 2 \implies r = \frac{2}{a} \]

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

### Step 5: Simplify the equation
Simplifying the denominator:
\[ 1 - \frac{2}{a} = \frac{a - 2}{a} \]
Now substituting this back into 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:
\[ a^2 = 8(a - 2) \]
Expanding the right side:
\[ a^2 = 8a - 16 \]

### Step 7: Rearrange the equation
Rearranging gives us a standard quadratic equation:
\[ a^2 - 8a + 16 = 0 \]

### Step 8: Factor the quadratic equation
This can be factored as:
\[ (a - 4)(a - 4) = 0 \]
Thus, we have:
\[ a - 4 = 0 \implies a = 4 \]

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