N2015 P1 Q6

Write down the first three non-zero terms in the Maclaurin series for \(\ln(1+2x)\), where \(-\frac12<x\le\frac12\), simplifying the coefficients.[2]

It is given that the three terms found in part (i) are equal to the first three terms in the series expansion of \(ax(1+bx)^c\) for small \(x\). Find the exact values of the constants \(a\), \(b\) and \(c\) and use these values to find the coefficient of \(x^4\) in the expansion of \(ax(1+bx)^c\), giving your answer as a simplified rational number.[6]