What is the number of terms in the expansion of
`(3 + 2 sqrt(5) )^(18) - (3-2 sqrt(5) )^(18)`

Answer

To find the number of terms in the expansion of \( (3 + 2\sqrt{5})^{18} - (3 - 2\sqrt{5})^{18} \), we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, we have two expansions to consider: \( (3 + 2\sqrt{5})^{18} \) and \( (3 - 2\sqrt{5})^{18} \).

### Step 2: Expand Both Terms
1. **First Expansion**:
\[ (3 + 2\sqrt{5})^{18} = \sum_{k=0}^{18} \binom{18}{k} (3)^{18-k} (2\sqrt{5})^k \]

2. **Second Expansion**:
\[ (3 - 2\sqrt{5})^{18} = \sum_{k=0}^{18} \binom{18}{k} (3)^{18-k} (-2\sqrt{5})^k \]

### Step 3: Combine the Expansions
Now, we need to consider the difference:
\[ (3 + 2\sqrt{5})^{18} - (3 - 2\sqrt{5})^{18} \]
When we subtract these two expansions, the even powers of \( k \) will cancel out because they will have opposite signs, while the odd powers will remain.

### Step 4: Identify Remaining Terms
The remaining terms will be those where \( k \) is odd. The odd values of \( k \) from 0 to 18 are:
1, 3, 5, 7, 9, 11, 13, 15, 17.

### Step 5: Count the Odd Terms
To find how many odd terms there are from 1 to 17:
- The odd integers in this range are: 1, 3, 5, 7, 9, 11, 13, 15, 17.
- This is an arithmetic sequence where the first term \( a = 1 \), the last term \( l = 17 \), and the common difference \( d = 2 \).

The number of terms \( n \) in an arithmetic sequence can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{17 - 1}{2} + 1 = \frac{16}{2} + 1 = 8 + 1 = 9 \]

### Conclusion
Thus, the number of terms in the expansion of \( (3 + 2\sqrt{5})^{18} - (3 - 2\sqrt{5})^{18} \) is **9**.