The first term of a geometric progression is twice its common ratio. Find the sum of the first two terms of the G.P if its sum to infinity is 8.
The correct answer is C. 72/25
Let the first term of the geometric progression be \(a\) and the common ratio be \(r\).
Given that the first term is twice the common ratio, we have:
\[a = 2r\]
The sum of an infinite geometric progression is given by the formula:
\[S_\infty = \frac{a}{1 - r}\]
Given that the sum to infinity is 8, we can write:
\[8 = \frac{2r}{1 - r}\]
Now, let's solve for \(r\):
\[8(1 - r) = 2r\]
\[8 - 8r = 2r\]
\[10r = 8\]
\[r = \frac{4}{5}\]
Substitute the value of \(r\) back into the equation \(a = 2r\):
\[a = 2\left(\frac{4}{5}\right) = \frac{8}{5}\]
The first two terms of the geometric progression are \(a\) and \(ar\):
First term: \(a = \frac{8}{5}\)
Second term: \(ar = \frac{8}{5} \times \frac{4}{5} = \frac{32}{25}\)
Now, find the sum of the first two terms:
\[S_2 = \frac{8}{5} + \frac{32}{25} = \frac{72}{25}\]
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