The first term of a G.P. is `-3` and the square of the second term is equal to its `4^(th)` term. Find its `7^(th)` term.

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

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

Let the common ratio be:
\[ r \]

### Step 2: Write the expressions for the second and fourth terms
The second term of the G.P. can be expressed as:
\[ \text{Second term} = a \cdot r = -3r \]

The fourth term of the G.P. can be expressed as:
\[ \text{Fourth term} = a \cdot r^3 = -3r^3 \]

### Step 3: Set up the equation based on the given condition
According to the problem, the square of the second term is equal to the fourth term:
\[ (\text{Second term})^2 = \text{Fourth term} \]
Thus, we have:
\[ (-3r)^2 = -3r^3 \]

### Step 4: Simplify the equation
Expanding the left side:
\[ 9r^2 = -3r^3 \]

Now, rearranging the equation gives:
\[ 3r^3 + 9r^2 = 0 \]

### Step 5: Factor the equation
Factoring out the common term:
\[ 3r^2(r + 3) = 0 \]

### Step 6: Solve for r
Setting each factor to zero gives us:
1. \( 3r^2 = 0 \) which implies \( r = 0 \) (not valid as it would make the G.P. trivial)
2. \( r + 3 = 0 \) which implies \( r = -3 \)

Thus, the common ratio is:
\[ r = -3 \]

### Step 7: Find the seventh term
The formula for the nth term of a G.P. is given by:
\[ T_n = a \cdot r^{n-1} \]
For the seventh term (\( n = 7 \)):
\[ T_7 = a \cdot r^{6} \]
Substituting the values of \( a \) and \( r \):
\[ T_7 = -3 \cdot (-3)^{6} \]

### Step 8: Calculate \( (-3)^{6} \)
Calculating \( (-3)^{6} \):
\[ (-3)^{6} = 729 \]

### Step 9: Calculate \( T_7 \)
Now substituting back:
\[ T_7 = -3 \cdot 729 = -2187 \]

### Final Answer
The seventh term of the G.P. is:
\[ T_7 = -2187 \]

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