The sum to infinity of a geometric progression with first term \(a\) and common ratio \(r\) is given by:

\(S = \frac{a}{1-r}\)

For the first progression, \(S = \frac{a}{1-r}\).

For the second progression, the sum to infinity is \(3S = \frac{a}{1-2r}\).

Equating the expressions for \(3S\):

\(3 \left( \frac{a}{1-r} \right) = \frac{a}{1-2r}\)

Cancel \(a\) from both sides (since \(a \neq 0\)):

\(\frac{3}{1-r} = \frac{1}{1-2r}\)

Cross-multiply to solve for \(r\):

\(3(1-2r) = 1-r\)

\(3 - 6r = 1 - r\)

Rearrange to find \(r\):

\(3 - 1 = 6r - r\)

\(2 = 5r\)

\(r = \frac{2}{5}\)