Find the next two terms of the series :
`2-6+18-54 . . . . . . . . . . . . .`

Answer

To find the next two terms of the series \(2, -6, 18, -54, \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term of the series is:
\[ a_1 = 2 \]
The second term is:
\[ a_2 = -6 \]
To find the common ratio \(r\), we can calculate:
\[ r = \frac{a_2}{a_1} = \frac{-6}{2} = -3 \]

### Step 2: Verify the common ratio with the next terms
Now, let's verify the common ratio with the next terms:
- For \(a_3\):
\[ r = \frac{a_3}{a_2} = \frac{18}{-6} = -3 \]
- For \(a_4\):
\[ r = \frac{a_4}{a_3} = \frac{-54}{18} = -3 \]

Since the common ratio \(r\) is consistently \(-3\) for all terms, we confirm that this series is a geometric progression (GP) with:
\[ a = 2 \quad \text{and} \quad r = -3 \]

### Step 3: Find the next two terms
To find the next terms \(a_5\) and \(a_6\), we can use the formula for the \(n\)-th term of a geometric progression:
\[ a_n = a \cdot r^{n-1} \]

#### Finding \(a_5\):
\[ a_5 = a \cdot r^{5-1} = 2 \cdot (-3)^4 \]
Calculating \((-3)^4\):
\[ (-3)^4 = 81 \]
Thus,
\[ a_5 = 2 \cdot 81 = 162 \]

#### Finding \(a_6\):
\[ a_6 = a \cdot r^{6-1} = 2 \cdot (-3)^5 \]
Calculating \((-3)^5\):
\[ (-3)^5 = -243 \]
Thus,
\[ a_6 = 2 \cdot (-243) = -486 \]

### Final Answer
The next two terms of the series are:
\[ 162, -486 \]