Final answer:
To determine the values of the constants a and b in the given expansion, we can compare the terms of the expansion to the general form given by the binomial theorem. By equating the coefficients of x in the first three terms, we can solve for a and b, which are found to be -2 and 9, respectively.
Step-by-step explanation:
To determine the values of the constants a and b in the expansion of (1-2x)² (1+ax)⁶ in ascending powers of x, we need to compare the first three terms of the expansion, which are 1 - x + bx², to the general form of the expansion given by the binomial theorem.
Using the binomial theorem, the general form of the expansion is given by (a + b)² (1 + ax)⁶. Comparing the terms of the expansion, we can see that the coefficient of x in the first term is -1, the coefficient of x in the second term is 1, and the coefficient of x in the third term is 0.0211.
So, equating the coefficients, we have:
-1 = -2a
1 = -a + b
0.0211 = -2a²
Solving these equations simultaneously, we find that a = -2 and b = 9.