Question

The 4th term of a G.P. is square of its second term, and the 1st term is -3. Determine its 7th term.

Hint:

For a G.P., use the formula \( T_n = a \cdot r^{n-1} \) and use the given relationships to solve for unknown terms.

Answer 1

Step 1: Define the general form of the G.P.
In a geometric progression (G.P.), the nth term is given by: \[ T_n = a \cdot r^{n-1} \] where \( a \) is the first term and \( r \) is the common ratio.
Step 2: Use the given information to form equations.
We are given that the 4th term is the square of the 2nd term: \[ T_4 = (T_2)^2 \] Substitute the formula for the nth term into this equation: \[ a \cdot r^3 = (a \cdot r)^2 \]
Step 3: Solve for the common ratio \( r \).
Simplify the equation: \[ a \cdot r^3 = a^2 \cdot r^2 \] Since \( a = -3 \), substitute it into the equation: \[ -3 \cdot r^3 = (-3)^2 \cdot r^2 \] \[ -3 \cdot r^3 = 9 \cdot r^2 \] \[ r = -3 \]
Step 4: Find the 7th term.
Now that we know \( a = -3 \) and \( r = -3 \), we can calculate the 7th term: \[ T_7 = a \cdot r^{6} = -3 \cdot (-3)^6 \] \[ T_7 = -3 \cdot 729 = -2187 \]