Bot GPT 4-o mini Let's denote the first term of the geometric progression (GP) as \( a \) and the common ratio as \( r \).
According to the problem, the first term \( a \) is two times its common ratio \( r \), which can be written as:
\[ a = 2r \]
The sum to infinity of a GP is given by the formula:
\[ S_{\infty} = \frac{a}{1 - r} \]
We are given that the sum to infinity is equal to 8:
\[ \frac{a}{1 - r} = 8 \]
Now, substituting \( a \) from the first equation into the sum to infinity equation:
\[ \frac{2r}{1 - r} = 8 \]
To solve for \( r \), we can cross-multiply:
\[ 2r = 8(1 - r) \]
Expanding the right side:
\[ 2r = 8 - 8r \]
Now, bringing all terms involving \( r \) to one side:
\[ 2r + 8r = 8 \]
\[ 10r = 8 \]
Dividing both sides by 10:
\[ r = \frac{8}{10} = 0.8 \]
Now that we have \( r \), we can find \( a \):
\[ a = 2r = 2 \times 0.8 = 1.6 \]
Now, we can find the sum of the first two terms of the GP. The first two terms are \( a \) and \( ar \):
- The first term \( a = 1.6 \).
- The second term \( ar = 1.6 \times 0.8 = 1.28 \).
Now, the sum of the first two terms \( S_2 \):
\[ S_2 = a + ar = 1.6 + 1.28 = 2.88 \]
Therefore, the sum of the first two terms of the progression is:
\[ \boxed{2.88} \]
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