Let's analyze and solve this question step-by-step.

Given Information

The expression $(1 + kx)^n$ is expanded in terms of powers of $x$ as: $(1 + kx)^n = 1 + Ax + 264x^2 + 1760x^3 + \dots$ We are required to:

1. Show that $(n - 2)k = 20$.
2. Determine the value of $A$.

Solution

1. Part (a): Showing that $(n - 2)k = 20$

   We can use the binomial expansion formula for $(1 + kx)^n$, which is: $(1 + kx)^n = \sum_{r=0}^{n} \binom{n}{r} (k x)^r = \sum_{r=0}^{n} \binom{n}{r} k^r x^r$

   Therefore, the coefficients of $x^r$ in the expansion are given by: $\text{Coefficient of } x^r = \binom{n}{r} k^r$

   We are given that the coefficient of $x^2$ is 264. So: $\binom{n}{2} k^2 = 264$

   The binomial coefficient $\binom{n}{2}$ is: $\binom{n}{2} = \frac{n(n-1)}{2}$

   Substituting this, we get: $\frac{n(n-1)}{2} k^2 = 264$ $n(n-1) k^2 = 528$

   We are also given that the coefficient of $x^3$ is 1760, so: $\binom{n}{3} k^3 = 1760$

   Using $\binom{n}{3} = \frac{n(n-1)(n-2)}{6}$, we substitute: $\frac{n(n-1)(n-2)}{6} k^3 = 1760$ $n(n-1)(n-2) k^3 = 10560$

   By dividing these two equations, we can solve for $k$ and check if $(n - 2)k = 20$ holds.

2. Part (b): Determining the value of $A$

   The coefficient $A$ represents the coefficient of $x$ in the expansion. Using the formula: $A = \binom{n}{1} k = n k$

   We substitute the values of $n$ and $k$ (after solving the previous equations) to find $A$.

Would you like a further detailed solution for each of these steps?

Related Questions:

1. What is the general formula for the binomial expansion of $(1 + kx)^n$?
2. How do we solve simultaneous equations involving binomial coefficients?
3. What would happen if $k$ or $n$ were negative in such expansions?
4. How can we find higher-order coefficients like $x^4$ in the expansion?
5. What are some practical applications of binomial expansion in physics or engineering?

Tip: Always verify each step when solving equations with binomial coefficients to ensure accuracy.