Unknown parameters and products

In \((1+kx)^n\), the coefficients of \(x\), \(x^2\) and \(x^3\) are \(nk\), \(\frac{n(n-1)}2k^2\) and \(\frac{n(n-1)(n-2)}6k^3\).

Write an equation for each given coefficient. Preserve zero cases before dividing by a parameter; apply any stated sign or integer restriction to the final answers.

For non-zero consecutive coefficients \(c_r\) of \((1+kx)^n\), \(\dfrac{c_{r+1}}{c_r}=\dfrac{n-r}{r+1}k\).

To obtain the coefficient of \(x^3\) in a product, combine the constant with the cubic term, the linear with the quadratic term, the quadratic with the linear term, and the cubic with the constant.

Only compute terms that can contribute to the required power. When negative powers occur, check the exponent combinations before truncating.