Find which term of G.P. `3-6+12-24+`. . . . . . . . . . . Is `-384` ?

Answer

To find which term of the geometric progression (G.P.) \(3, -6, 12, -24, \ldots\) is \(-384\), we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \(a\) of the G.P. is:
\[ a = 3 \]

To find the common ratio \(r\), we can use the formula:
\[ r = \frac{t_2}{t_1} = \frac{-6}{3} = -2 \]

### Step 2: Use the formula for the nth term of a G.P.
The formula for the nth term \(a_n\) of a G.P. is given by:
\[ a_n = a \cdot r^{n-1} \]
We need to find \(n\) such that \(a_n = -384\).

### Step 3: Set up the equation
Substituting the known values into the nth term formula:
\[ -384 = 3 \cdot (-2)^{n-1} \]

### Step 4: Simplify the equation
First, divide both sides by 3:
\[ -128 = (-2)^{n-1} \]

### Step 5: Express \(-128\) as a power of \(-2\)
We know that:
\[ -128 = -2^7 \]
Thus, we can write:
\[ (-2)^{n-1} = -2^7 \]

### Step 6: Compare the exponents
Since the bases are the same, we can equate the exponents:
\[ n - 1 = 7 \]

### Step 7: Solve for \(n\)
Adding 1 to both sides gives:
\[ n = 8 \]

### Conclusion
The term \(-384\) is the 8th term of the G.P.

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