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### Problem (i)
You need to find the first three terms in the binomial expansion of \( (8 - 9x)^6 \) in ascending powers of \( x \).

### Problem (ii)
You need to state the set of values of \( x \) for which this expansion is valid.

### Solution to Part (i):
The binomial theorem states that for any \( (a + b)^n \), the expansion is given by:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

For \( (8 - 9x)^6 \), we can treat \( a = 8 \) and \( b = -9x \), and \( n = 6 \).

The first three terms will be:

1. **First term (k=0):**
   \[
   \text{First term} = \binom{6}{0} \cdot 8^{6-0} \cdot (-9x)^0 = 1 \cdot 8^6 \cdot 1 = 262144
   \]

2. **Second term (k=1):**
   \[
   \text{Second term} = \binom{6}{1} \cdot 8^{6-1} \cdot (-9x)^1 = 6 \cdot 8^5 \cdot (-9x) = -884736x
   \]

3. **Third term (k=2):**
   \[
   \text{Third term} = \binom{6}{2} \cdot 8^{6-2} \cdot (-9x)^2 = 15 \cdot 8^4 \cdot 81x^2 = 1244160x^2
   \]

So, the first three terms are:

\[
262144 - 884736x + 1244160x^2
\]

### Solution to Part (ii):
The binomial expansion of \( (1 + u)^n \) is valid when \( |u| < 1 \).

For the expression \( (8 - 9x)^6 \) to be valid, we can rewrite it in the form \( 8(1 - \frac{9x}{8})^6 \). The expression \( (1 - \frac{9x}{8})^6 \) is valid for:

\[
\left|\frac{9x}{8}\right| < 1 \quad \Rightarrow \quad |x| < \frac{8}{9}
\]

Thus, the set of values of \( x \) for which this expansion is valid is:

\[
|x| < \frac{8}{9}
\]

### Summary:
- **First three terms:** \( 262144 - 884736x + 1244160x^2 \)
- **Valid values of \( x \):** \( |x| < \frac{8}{9} \)

Would you like further details on any part of this problem or explanations for each step?

### Related Questions:
1. How would the first three terms change if the exponent were 7 instead of 6?
2. What would the expansion be if the binomial were \( (2 - 3x)^6 \)?
3. How do you derive the general term in the binomial expansion?
4. What are the conditions for a binomial series to converge?
5. Can you explain how the binomial expansion relates to the Taylor series?

### Tip:
For binomial expansions involving negative or fractional powers, the expansion can represent an infinite series, and the radius of convergence becomes essential.

Find the first three terms in the binomial expansion of (8 - 9x)^6 in ascending powers of x. State the set of values of x for which this expansion is valid.

Learn how to find the first three terms in the binomial expansion of (8 - 9x)^6 and determine the set of values for x for which this expansion is valid. This problem covers advanced algebraic concepts and the application of the binomial theorem.

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