Write down the sequence whose `n ^(th)` term is `( (-2) ^(n))/( (-1) ^(n) +2)`

Answer

To find the sequence whose \( n^{th} \) term is given by

\[ a_n = \frac{(-2)^n}{(-1)^n + 2} \]

we will calculate the first few terms of the sequence by substituting \( n = 1, 2, 3, 4, \) and \( 5 \).

### Step 1: Calculate the first term \( a_1 \)

Substituting \( n = 1 \):

\[ a_1 = \frac{(-2)^1}{(-1)^1 + 2} = \frac{-2}{-1 + 2} = \frac{-2}{1} = -2 \]

### Step 2: Calculate the second term \( a_2 \)

Substituting \( n = 2 \):

\[ a_2 = \frac{(-2)^2}{(-1)^2 + 2} = \frac{4}{1 + 2} = \frac{4}{3} \]

### Step 3: Calculate the third term \( a_3 \)

Substituting \( n = 3 \):

\[ a_3 = \frac{(-2)^3}{(-1)^3 + 2} = \frac{-8}{-1 + 2} = \frac{-8}{1} = -8 \]

### Step 4: Calculate the fourth term \( a_4 \)

Substituting \( n = 4 \):

\[ a_4 = \frac{(-2)^4}{(-1)^4 + 2} = \frac{16}{1 + 2} = \frac{16}{3} \]

### Step 5: Calculate the fifth term \( a_5 \)

Substituting \( n = 5 \):

\[ a_5 = \frac{(-2)^5}{(-1)^5 + 2} = \frac{-32}{-1 + 2} = \frac{-32}{1} = -32 \]

### Summary of the first five terms

Now we can summarize the first five terms of the sequence:

- \( a_1 = -2 \)
- \( a_2 = \frac{4}{3} \)
- \( a_3 = -8 \)
- \( a_4 = \frac{16}{3} \)
- \( a_5 = -32 \)

### Final Sequence

The sequence can be written as:

\[ -2, \frac{4}{3}, -8, \frac{16}{3}, -32, \ldots \]