Let A1, G1, H1 denote the arithmetic, geometric and harmoni c means, respectively, of two distinct positive numbers. For n ≥ 2, Let An – 1 and Hn – 1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectivelyWhich one of the following statements is correct ?a)G1 G2 G3 ........b)G1

Question

Let A1, G1, H1 denote the arithmetic, geometric and harmoni c means, respectively, of two distinct positive numbers. For n ≥ 2, Let An – 1 and Hn – 1 have arithmetic, geometric and harmonic means as An, Gn, Hn respectivelyWhich one of the following statements is correct ?a)G1 > G2 > G3 > ........b)G1 < G2 < G3 < ........c)G1 = G2 = G3 = ........d)G1 < G3 < G5 < ...... and G2 > G4 > G6 > ......Correct answer is option 'C'. Can you explain this answer?

Answer

Let's first find the formulas for the arithmetic, geometric, and harmonic means of two numbers.

Arithmetic Mean (A):
The arithmetic mean of two numbers a and b is simply the average of the two numbers, which can be calculated using the formula:
A = (a + b) / 2

Geometric Mean (G):
The geometric mean of two numbers a and b is the square root of their product, which can be calculated using the formula:
G = sqrt(a * b)

Harmonic Mean (H):
The harmonic mean of two numbers a and b is the reciprocal of the average of their reciprocals, which can be calculated using the formula:
H = 2 / (1/a + 1/b)

Now, let's substitute n as the arithmetic mean (A1) of two positive numbers a and b and try to find the values of the geometric mean (G1) and harmonic mean (H1) in terms of n.

Given: A1 = n

Substituting this value in the formulas for G and H:

G1 = sqrt(a * b)
H1 = 2 / (1/a + 1/b)

We need to find the values of G1 and H1 in terms of n.

To find G1, we can express a and b in terms of n using the formula for A:
a = n + k
b = n - k

where k is the difference between a and n (or b and n).

Substituting these values in the formula for G:

G1 = sqrt((n + k) * (n - k))

Now, let's simplify G1:
G1 = sqrt(n^2 - k^2)

Similarly, to find H1, we can express a and b in terms of n using the formula for A:
a = n + m
b = n - m

where m is the difference between a and n (or b and n).

Substituting these values in the formula for H:

H1 = 2 / (1/(n + m) + 1/(n - m))

Now, let's simplify H1:
H1 = 2 / ((2n) / (n^2 - m^2))

H1 = 2(n^2 - m^2) / (2n)

H1 = (n^2 - m^2) / n

So, the formulas for the geometric mean (G1) and harmonic mean (H1) in terms of the arithmetic mean (A1 = n) are:

G1 = sqrt(n^2 - k^2)
H1 = (n^2 - m^2) / n

Similar JEE Doubts

Let A1, G1, H1 denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For n ≥ 2, let An−1 and Hn−1 has arithmetic, geometric and harmonic means as An, Gn, Hn respectively.Which of the following statements is correct?

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