Write the first four terms of the sequence whose nth term is given
`sin^(n) 30^(@)`

Answer

To find the first four terms of the sequence whose nth term is given by \( \sin^n(30^\circ) \), we will follow these steps:

### Step 1: Calculate the first term
For \( n = 1 \):
\[ \text{First term} = \sin^1(30^\circ) = \sin(30^\circ) \]
Using the known value:
\[ \sin(30^\circ) = \frac{1}{2} \]
So, the first term is:
\[ \frac{1}{2} \]

### Step 2: Calculate the second term
For \( n = 2 \):
\[ \text{Second term} = \sin^2(30^\circ) = \left(\sin(30^\circ)\right)^2 = \left(\frac{1}{2}\right)^2 \]
Calculating this gives:
\[ \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \]

### Step 3: Calculate the third term
For \( n = 3 \):
\[ \text{Third term} = \sin^3(30^\circ) = \left(\sin(30^\circ)\right)^3 = \left(\frac{1}{2}\right)^3 \]
Calculating this gives:
\[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8} \]

### Step 4: Calculate the fourth term
For \( n = 4 \):
\[ \text{Fourth term} = \sin^4(30^\circ) = \left(\sin(30^\circ)\right)^4 = \left(\frac{1}{2}\right)^4 \]
Calculating this gives:
\[ \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{16} \]

### Final Result
The first four terms of the sequence are:
\[ \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \]