JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Sequences and Series

Let

A_{1}, G_{1}, H_{1}

denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For

n \geq 2

, let

A_{n - 1}

and

H_{n - 1}

has arithmetic, geometric and harmonic means as

A_{n}, G_{n}, H_{n}

respectively.

Which one of the following statements is correct?

Answer: (B)

Solution

For two distinct positive numbers

a

and

b

A_{1} = \frac{a + b}{2}, G_{1} = ab, H_{1} = \frac{2 ab}{a + b}

For

n \geq 2

G_{n} = A_{n - 1} \cdot H_{n - 1}

Since

A_{n - 1}

and

H_{n - 1}

are the arithmetic and harmonic means of

A_{n - 2}

and

H_{n - 2}

, and the geometric mean of two numbers is the geometric mean of their AM and HM:

G_{n} = A_{n - 1} \cdot H_{n - 1} = ab \cdot ab = ab = G_{1}

Therefore,

G_{1} = G_{2} = G_{3} = \dots = ab

JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Single correct option