Given that the binomial expansion of (1 + kx) ^ n is 1 - 6x + 30x^2 + ..., find the values of n and k.
Setting (1 + kx)n equal to 1 - 6x + 30x2 + ..., the binomial expansion can be applied to the LHS, by making use of the formula provided in the formula book. In this case, where x appears in the formula book expression, we must replace it by kx. This yields:1 + n(kx) + n(n-1)(kx)/2! + ... = 1 - 6x + 30x2 + ...Since LHS = RHS, we can equate the coefficients of x and x2 to give two equations in n and k:x: nk = -6x2: n(n-1)k/2 = 30This pair of simultaneous equations are most easily solved by noting the term nk appears in the second equation, and can be substituted for the value -6 as given by the first equation. This gives a linear equation in terms of n, which can be solved to give n = -9. Noting the question asks for a value of k as well, we substitute n = -9 into the first equation to give k = -2/3.
