Find 5 geometric means between 1 and 27.

Let a, b, c and d be the 5 geometric means 1 and 27.

∴ 1, a, b, c, d, e and 27 are in G.P.

So, first term (a) = 1 and seventh term = 27.

Let common ratio be r.

∴ ar⁶ = 27

⇒ (1)r⁶ = 27

⇒ r⁶ = 3³

⇒ r⁶ = $(\sqrt{3})^6$

⇒ r = $\sqrt{3}$.

Second term = ar = $1 \times \sqrt{3} = \sqrt{3}$,

Third term = ar² = $1 \times (\sqrt{3})^2 = 3$,

Fourth term = ar³ = $1 \times (\sqrt{3})^3 = 3\sqrt{3}$,

Fifth term = ar⁴ = $1 \times (\sqrt{3})^4 = 9$,

Sixth term = ar⁵ = $1 \times (\sqrt{3})^5 = 9\sqrt{3}$.

Hence, 5 geometric means between 1 and 27 = $\sqrt{3}, 3, 3\sqrt{3}, 9, 9\sqrt{3}$.