Question

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

Please full steps, without any guess (Don't answer, if you're not sure)

Answer 1

GP.

First term is a. a = -3

nth term of GP = arⁿ ⁻ ¹

The 4th term = ar⁴ ⁻ ¹ = ar³

The 2nd term = ar² ⁻ ¹ = ar

4th = square of 2nd

ar³ = (ar)²

ar³ = a²r²

Divide both sides by ar²

ar³ / ar² = a²r²/ar²

r = a

Therefore a = -3, r = -3

7th term = ar⁷ ⁻ ¹ = ar⁶ = (-3)(-3)⁶ = (-3)*(-3)*(-3)*(-3)*(-3)*(-3)*(-3) = -2187

The 7th term = -2187

Hope this helps.

Answer 2

Let, First term of the G.P. = a = -3
Common ratio of the G.P. = r

It is known that, aₓ = arˣ⁻¹
a₄ = ar³ = (-3) r³
a₂ = ar¹ = (-3) r

Now, According to the given condition,
⇒ (-3) r³ = [(-3) r]²
⇒ -3r³ = 9 r²
⇒ r = -3

Now, a₇ = a r⁷⁻¹ = a r⁶ = (-3) (-3)⁶ = (-3)⁷ = -2187

Thus, the seventh term of the G.P. is -2187

Hope this helps!