(i) The first two terms of the expansion of \((1 + ax)^{\frac{2}{3}}\) are \(1 + \frac{2}{3}ax\). The expression for the coefficient of \(x\) in the expansion of \((1 + 2x)(1 + ax)^{\frac{2}{3}}\) is:

\(2 \cdot 1 + 1 \cdot \frac{2}{3}a = 0\)

Solving for \(a\):

\(2 + \frac{2}{3}a = 0\)

\(\frac{2}{3}a = -2\)

\(a = -3\)

(ii) Substitute \(a = -3\) into \((1 + ax)^{\frac{2}{3}}\) to get \((1 - 3x)^{\frac{2}{3}}\). The unsimplified terms in \(x^2\) and \(x^3\) are obtained from the expansion:

\((1 - 3x)^{\frac{2}{3}} = 1 - 2x + \frac{10}{3}x^2 - \frac{20}{9}x^3 + \ldots\)

Multiply by \(1 + 2x\) to find the term in \(x^3\):

\((1 + 2x)(1 - 2x + \frac{10}{3}x^2 - \frac{20}{9}x^3)\)

The term in \(x^3\) is:

\(-2 \cdot \frac{10}{3} + 1 \cdot \left(-\frac{20}{9}\right) = -\frac{20}{3} - \frac{20}{9}\)

Combine the fractions:

\(-\frac{60}{9} - \frac{20}{9} = -\frac{80}{9}\)

Thus, the term in \(x^3\) is \(-\frac{10}{3}x^3\).