The 4th term of a G.P. is square of its second term, and the first term is -3. Determine its 7th term.

Explanation

To solve the problem step by step, we need to determine the 7th term of a geometric progression (G.P.) given that the 4th term is the square of the 2nd term and the first term is -3.

### Step 1: Define the terms of the G.P.
Let:
- The first term \( a = -3 \)
- The common ratio \( r \)

The terms of the G.P. can be expressed as:
- 1st term: \( a = -3 \)
- 2nd term: \( ar = -3r \)
- 3rd term: \( ar^2 = -3r^2 \)
- 4th term: \( ar^3 = -3r^3 \)

### Step 2: Set up the equation based on the given condition
According to the problem, the 4th term is equal to the square of the 2nd term:
\[ ar^3 = (ar)^2 \]
Substituting the expressions for the terms:
\[ -3r^3 = (-3r)^2 \]
This simplifies to:
\[ -3r^3 = 9r^2 \]

### Step 3: Rearrange the equation
To solve for \( r \), rearrange the equation:
\[ -3r^3 - 9r^2 = 0 \]
Factoring out common terms:
\[ -3r^2(r + 3) = 0 \]

### Step 4: Solve for \( r \)
Setting each factor to zero gives:
1. \( -3r^2 = 0 \) → \( r = 0 \) (not valid for G.P.)
2. \( r + 3 = 0 \) → \( r = -3 \)

### Step 5: Find the 7th term
Now that we have \( r = -3 \), we can find the 7th term:
\[ a_7 = ar^6 \]
Substituting the values:
\[ a_7 = -3(-3)^6 \]

### Step 6: Calculate \( (-3)^6 \)
Calculating \( (-3)^6 \):
\[ (-3)^6 = 729 \]
Thus,
\[ a_7 = -3 \times 729 = -2187 \]

### Final Answer
The 7th term of the G.P. is:
\[ \boxed{-2187} \]