Find the number of terms in the following expansions.
(vi) `(1+ 3 sqrt(5) x )^(9) - (1-3 sqrt(5) x)^(9)`

Answer

To find the number of terms in the expansion of \((1 + 3\sqrt{5}x)^9 - (1 - 3\sqrt{5}x)^9\), we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \((a + b)^n\) is given by:
\[ \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, we will expand both \((1 + 3\sqrt{5}x)^9\) and \((1 - 3\sqrt{5}x)^9\).

### Step 2: Expand \((1 + 3\sqrt{5}x)^9\)
Using the binomial theorem, we expand:
\[ (1 + 3\sqrt{5}x)^9 = \sum_{k=0}^{9} \binom{9}{k} (3\sqrt{5}x)^k \]
This gives us terms of the form:
\[ \binom{9}{k} (3\sqrt{5})^k x^k \]

### Step 3: Expand \((1 - 3\sqrt{5}x)^9\)
Similarly, we expand:
\[ (1 - 3\sqrt{5}x)^9 = \sum_{k=0}^{9} \binom{9}{k} (-3\sqrt{5}x)^k \]
This gives us terms of the form:
\[ \binom{9}{k} (-3\sqrt{5})^k x^k \]

### Step 4: Subtract the Two Expansions
Now, we subtract the two expansions:
\[ (1 + 3\sqrt{5}x)^9 - (1 - 3\sqrt{5}x)^9 \]
When we perform the subtraction, the even powers of \(x\) will cancel out because they will have opposite signs, while the odd powers will remain.

### Step 5: Identify the Remaining Terms
The remaining terms will be those with odd powers of \(x\):
- The odd powers from \(k = 1, 3, 5, 7, 9\) will remain.
- The terms are:
- For \(k=1\): \( \binom{9}{1} (3\sqrt{5})^1 x^1 \)
- For \(k=3\): \( \binom{9}{3} (3\sqrt{5})^3 x^3 \)
- For \(k=5\): \( \binom{9}{5} (3\sqrt{5})^5 x^5 \)
- For \(k=7\): \( \binom{9}{7} (3\sqrt{5})^7 x^7 \)
- For \(k=9\): \( \binom{9}{9} (3\sqrt{5})^9 x^9 \)

### Step 6: Count the Remaining Terms
The remaining terms correspond to the odd values of \(k\) from 1 to 9:
- \(k = 1\)
- \(k = 3\)
- \(k = 5\)
- \(k = 7\)
- \(k = 9\)

Thus, there are a total of **5 terms** remaining in the expansion.

### Final Answer
The number of terms in the expansion of \((1 + 3\sqrt{5}x)^9 - (1 - 3\sqrt{5}x)^9\) is **5**.

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