Find the number of terms in the following expansions.
(ii) `( 5x-(1)/(x^3) )^(17)`

Answer

To find the number of terms in the expansion of \( (5x - \frac{1}{x^3})^{17} \), we can follow these steps:

### Step 1: Identify the general form
The expression can be written as \( (a + b)^n \) where:
- \( a = 5x \)
- \( b = -\frac{1}{x^3} \)
- \( n = 17 \)

### Step 2: Use the Binomial Theorem
According to the Binomial Theorem, the expansion of \( (a + b)^n \) contains \( n + 1 \) terms. Therefore, the number of terms in the expansion is given by:
\[ \text{Number of terms} = n + 1 \]

### Step 3: Calculate the number of terms
Substituting the value of \( n \):
\[ \text{Number of terms} = 17 + 1 = 18 \]

### Step 4: Consider the powers of \( x \)
Next, we need to ensure that the terms are distinct. The general term in the expansion can be expressed as:
\[ T_k = \binom{n}{k} (5x)^{n-k} \left(-\frac{1}{x^3}\right)^k \]
This simplifies to:
\[ T_k = \binom{17}{k} (5^{n-k}) x^{(n-k) - 3k} \]
\[ T_k = \binom{17}{k} (5^{17-k}) x^{17 - 4k} \]

### Step 5: Determine the range of \( k \)
The exponent of \( x \) in each term is \( 17 - 4k \). To find the distinct powers of \( x \), we need to find the values of \( k \) that give different powers of \( x \).

### Step 6: Find the limits for \( k \)
The exponent \( 17 - 4k \) will be distinct for different values of \( k \) as long as \( k \) is within the limits:
- The minimum value of \( k \) is \( 0 \) (giving \( 17 - 4(0) = 17 \)).
- The maximum value of \( k \) is \( 4 \) (since \( 17 - 4k \) must be non-negative).

### Step 7: Calculate the distinct powers of \( x \)
The values of \( k \) can be \( 0, 1, 2, 3, 4 \). This gives us:
- For \( k = 0 \): \( x^{17} \)
- For \( k = 1 \): \( x^{13} \)
- For \( k = 2 \): \( x^{9} \)
- For \( k = 3 \): \( x^{5} \)
- For \( k = 4 \): \( x^{1} \)

### Step 8: Count the distinct terms
The distinct powers of \( x \) are \( 17, 13, 9, 5, 1 \), which gives us a total of \( 5 \) distinct terms.

### Final Result
Thus, the number of terms in the expansion of \( (5x - \frac{1}{x^3})^{17} \) is \( 5 \).

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