The `6^(th)` term in the expansion of `5^x` is

Options

A. `x^5/(5!)`
B. `x^4/(4!)`
C. `(x^5)/(5!) (log _e 5)^5`
D. `(x^5)/(5) (log _e 5)^5`

Answer

To find the sixth term in the expansion of \(5^x\), we will use the formula for the exponential series. The expansion of \(a^x\) can be expressed as:

\[ a^x = \sum_{n=0}^{\infty} \frac{(x \ln a)^n}{n!} \]

In our case, \(a = 5\). Therefore, we can rewrite the expansion as:

\[ 5^x = \sum_{n=0}^{\infty} \frac{(x \ln 5)^n}{n!} \]

### Step 1: Identify the general term
The general term \(T_n\) in the expansion is given by:

\[ T_n = \frac{(x \ln 5)^n}{n!} \]

### Step 2: Find the sixth term
To find the sixth term, we need to calculate \(T_5\) (since we start counting from \(T_0\)). Thus,

\[ T_5 = \frac{(x \ln 5)^5}{5!} \]

### Step 3: Simplify the expression
Now, we can simplify this expression:

\[ T_5 = \frac{x^5 (\ln 5)^5}{5!} \]

### Conclusion
Thus, the sixth term in the expansion of \(5^x\) is:

\[ \frac{x^5 (\ln 5)^5}{5!} \]

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