Number of terms in expansion of [(x + a) ^ n + (x - a) ^ 8] * i s\ =

To find the number of terms in the expansion of the given expression, let's carefully analyze each part of it.

### Given Expression
\[
\left[(x + a)^n + (x - a)^8\right] \times i
\]

We will break down the expression and determine the number of terms contributed by each part.

### Step-by-Step Solution

1. **Expansion of \((x + a)^n\):**

   The binomial expansion of \((x + a)^n\) is:
   \[
   (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k
   \]
   This expansion has \(n + 1\) terms (from \(k = 0\) to \(k = n\)).

2. **Expansion of \((x - a)^8\):**

   Similarly, the binomial expansion of \((x - a)^8\) is:
   \[
   (x - a)^8 = \sum_{m=0}^{8} \binom{8}{m} x^{8-m} (-a)^m
   \]
   This expansion has \(8 + 1 = 9\) terms (from \(m = 0\) to \(m = 8\)).

3. **Combining the expansions:**

   The expression inside the brackets is the sum of \((x + a)^n\) and \((x - a)^8\). To determine the number of distinct terms in this sum, we consider the exponents of \(x\) and \(a\) in both expansions.

   Since the powers of \(x\) in \((x + a)^n\) range from \(n\) down to \(0\) and in \((x - a)^8\) range from \(8\) down to \(0\), the range of the powers of \(x\) in the combined expression is from the minimum to the maximum of these ranges. 

   The range of powers of \(x\) will thus be from \(\min(0, 0) = 0\) to \(\max(n, 8)\). 

   Similarly, the powers of \(a\) in \((x + a)^n\) range from \(0\) to \(n\), and in \((x - a)^8\) from \(0\) to \(8\).

4. **Determining the number of distinct terms:**

   To find the number of distinct terms after the addition of the two expansions, consider all unique combinations of \(x^p a^q\) where \(p\) is the power of \(x\) and \(q\) is the power of \(a\).

   The maximum power of \(x\) is \(\max(n, 8)\), and the minimum power of \(x\) is \(0\).

   The maximum power of \(a\) is also \(\max(n, 8)\), and the minimum power of \(a\) is \(0\).

   Thus, the distinct terms in the combined expansion will have different combinations of powers of \(x\) and \(a\) ranging independently from \(0\) to \(\max(n, 8)\).

5. **Final Calculation of the Number of Terms:**

   Since \(x\) and \(a\) can independently range from \(0\) to \(\max(n, 8)\), there are \((\max(n, 8) + 1)^2\) distinct terms.

### Conclusion

The number of terms in the expansion of the expression \(\left[(x + a)^n + (x - a)^8\right] \times i\) is:

\[
(\max(n, 8) + 1)^2.
\]

This formula calculates the number of distinct combinations of the terms after expanding the given expression.

### Do you want more details, or do you have any questions?

### Related Questions:
1. What is the binomial theorem, and how does it help in expanding expressions like \((x + a)^n\)?
2. How would the number of terms change if we considered \((x - a)^n\) instead of \((x - a)^8\)?
3. Can we determine the coefficient of a specific term in the expansion? How?
4. How does the imaginary unit \(i\) affect the expansion in this context?
5. What are the possible terms in the expansion if \(n = 8\)?

**Tip:** When expanding binomials, remember that each term in the expansion represents a combination of the powers of each component in the binomial raised to the corresponding binomial coefficient.

Explore the number of terms in the expansion of [(x + a)^n + (x - a)^8] * i with a detailed mathematical analysis. This problem involves binomial expansions, powers and exponents, and combinations of terms. Suitable for advanced high school students and beyond.

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