\(a_n = a_1 r^{n-1}\)

\(a_6 = a_1 r^{6-1}\)

\(4 = a_1 r^5\)

\(a_40 = a_1 r^{10-1}\)

\(a_10 = a_1 r^{10-1}\)

\(\frac{4}{81} = a_1 r^9\)

Divide the two equations to eliminate \(a_1\)

\(\frac{4}{81(4)} = \frac{a_1r^9}{a_1r^5}\)

\(\frac{1}{81} = r^4\)

\(r=\frac{1}{3}\) Answer