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 the last term is(a) 1000 (b) 100 (c) \(\frac{1}{100}\)(d) \(\frac{1}{1000}\)
Answer» (c) \(\frac{1}{100}.\) Let the six numbers in G.P. be \(\frac{a}{r^5}\), \(\frac{a}{r^3}\), \(\frac{a}{r}\), ar, ar³, ar⁵ given, \(\frac{a}{r^5}\) x \(\frac{a}{r^3}\) x \(\frac{a}{r}\) x ar x ar³ x ar⁵= 1000 ⇒ a⁶ = 1000 ⇒ a = \(\sqrt{10}\) Given, T₄ = ar = 1 ⇒ \(\sqrt{10}\)r = 1 ⇒ = \(\frac{1}{\sqrt{10}}\) ∴ Last term of G.P. = ar⁵ = \(\sqrt{10}\) x \(\frac{1}{\sqrt{(10)}^5}\) = \(\frac{1}{100}.\)

Six positive numbers are in G.P, such that their product is 1000. If the fourth term is 1, then the last term is(a) 1000 (b) 100 (c) \(\frac{1}{100}\)(d) \(\frac{1}{1000}\)

Answer»

(c) \(\frac{1}{100}.\)

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

given, \(\frac{a}{r^5}\) x \(\frac{a}{r^3}\) x \(\frac{a}{r}\) x ar x ar³ x ar⁵= 1000

⇒ a⁶ = 1000 ⇒ a = \(\sqrt{10}\)

Given, T₄ = ar = 1 ⇒ \(\sqrt{10}\)r = 1 ⇒ = \(\frac{1}{\sqrt{10}}\)

∴ Last term of G.P. = ar⁵ = \(\sqrt{10}\) x \(\frac{1}{\sqrt{(10)}^5}\) = \(\frac{1}{100}.\)