Find the 22nd term of the G.P. -2,2,-2,… :

Options

A. `-22`
B. `-2`
C. 2
D. none of these

Answer

To find the 22nd term of the given geometric progression (G.P.) -2, 2, -2, ..., we can follow these steps:

### Step 1: Identify the first term (A) and the common ratio (R)
The first term \( A \) of the G.P. is:
\[ A = -2 \]

To find the common ratio \( R \), we can divide the second term by the first term:
\[ R = \frac{T_2}{T_1} = \frac{2}{-2} = -1 \]

### Step 2: Use the formula for the n-th term of a G.P.
The formula for the n-th term \( T_n \) of a G.P. is given by:
\[ T_n = A \cdot R^{(n-1)} \]

### Step 3: Substitute the values into the formula
We want to find the 22nd term, so we set \( n = 22 \):
\[ T_{22} = A \cdot R^{(22-1)} \]
\[ T_{22} = -2 \cdot (-1)^{21} \]

### Step 4: Calculate \( (-1)^{21} \)
Since 21 is an odd number, we have:
\[ (-1)^{21} = -1 \]

### Step 5: Substitute back to find \( T_{22} \)
Now substituting this value back into the equation:
\[ T_{22} = -2 \cdot (-1) \]
\[ T_{22} = 2 \]

### Final Answer
The 22nd term of the G.P. is:
\[ T_{22} = 2 \]

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