The sum of the first three terms of a geometric progression is half its sum to infinity. Find the positive common ratio of the progression.
Let the first term of the geometric progression (GP) be \( a \) and the common ratio be \( r \). The sum of the first three terms is: a + ar + ar\(^2\) = a(1 + r + r\(^2\))
\(S_3 = a(1 + r + r^2)\)
The sum to infinity is:
\(S_{\infty} = \frac{a}{1 - r}\)
But: \(S_3 = \frac{1}{2} S_{\infty}\)
Substituting the expressions gives:
\(a(1 + r + r^2) = \frac{1}{2} \cdot \frac{a}{1 - r}\)
Dividing by \( a \):
\(1 + r + r^2 = \frac{1}{2(1 - r)}\)
Cross-multiplying:
\(2(1 + r + r^2)(1 - r) = 1\)
Expanding and simplifying leads to:
\(2(1 - r^3) = 1\)
Thus: \(r^3 = \frac{1}{2}\)
Taking the cube root:
\(r = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}\)
Therefore, the positive common ratio of the progression is:
\(\frac{1}{\sqrt[3]{2}}\).
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