Six positive numbers are in G.P. such that their product is 1000. If t

1.	Six positive numbers are in G.P. such that their product is 1000. If the fourth term is 1, then find the last term.
Answer» Let the six numbers in G.P. be \(\frac{a}{r^5}\), \(\frac{a}{r^3}\), \(\frac{a}{r}\), ar, ar³, ar⁵. Then, Product = 1000 ⇒ \(\frac{a}{r^5}\) x \(\frac{a}{r^3}\) x \(\frac{a}{r}\) x ar x ar³ x ar⁵ ⇒ a6 = 1000 ⇒ a = \(\sqrt{10}\) Given, Fourth term = t₄ = ar = 1 ⇒ \(\sqrt{10}\) x r = 1 ⇒ \(\frac{1}{\sqrt{10}}\) ∴ Last term = ar⁵ = \(\sqrt{10}\) x\(\bigg(\)\(\frac{1}{\sqrt{10}}\)\(\bigg)\)= \(\frac{1}{100}.\)

Six positive numbers are in G.P. such that their product is 1000. If the fourth term is 1, then find the last term.

Answer»

Let the six numbers in G.P. be \(\frac{a}{r^5}\), \(\frac{a}{r^3}\), \(\frac{a}{r}\), ar, ar³, ar⁵.

Then, Product = 1000

⇒ \(\frac{a}{r^5}\) x \(\frac{a}{r^3}\) x \(\frac{a}{r}\) x ar x ar³ x ar⁵

⇒ a6 = 1000 ⇒ a = \(\sqrt{10}\)

Given, Fourth term = t₄ = ar = 1

⇒ \(\sqrt{10}\) x r = 1 ⇒ \(\frac{1}{\sqrt{10}}\)

∴ Last term = ar⁵ = \(\sqrt{10}\) x\(\bigg(\)\(\frac{1}{\sqrt{10}}\)\(\bigg)\)= \(\frac{1}{100}.\)