for a set of non zero five positive values are 8,10,12,14,16 prove that AM>=GM>=HM where AM,GM,HM are arithmetic mean,geometric mean and harmonic mean

1. Break Down the Problem

We need to prove the following inequalities for the given set of positive values:

• Arithmetic Mean (AM) $\geq$ Geometric Mean (GM)
• Geometric Mean (GM) $\geq$ Harmonic Mean (HM)

Let's denote the five values as $a_1 = 8$, $a_2 = 10$, $a_3 = 12$, $a_4 = 14$, $a_5 = 16$.

2. Relevant Concepts

• The Arithmetic Mean (AM) is given by: $\text{AM} = \frac{a_1 + a_2 + a_3 + a_4 + a_5}{5}$

• The Geometric Mean (GM) is given by: $\text{GM} = \sqrt[5]{a_1 \times a_2 \times a_3 \times a_4 \times a_5}$

• The Harmonic Mean (HM) is given by: $\text{HM} = \frac{5}{\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3} + \frac{1}{a_4} + \frac{1}{a_5}}$

3. Analysis and Detail

Step 1: Calculate AM

$\text{AM} = \frac{8 + 10 + 12 + 14 + 16}{5} = \frac{60}{5} = 12$

Step 2: Calculate GM

$\text{GM} = \sqrt[5]{8 \times 10 \times 12 \times 14 \times 16}$

Calculating the product: $8 \times 10 = 80$ $80 \times 12 = 960$ $960 \times 14 = 13440$ $13440 \times 16 = 215040$ Calculating the fifth root: $\text{GM} = \sqrt[5]{215040} \approx 11.458$

Step 3: Calculate HM

$\text{HM} = \frac{5}{\frac{1}{8} + \frac{1}{10} + \frac{1}{12} + \frac{1}{14} + \frac{1}{16}}$ Calculating the reciprocals: $\frac{1}{8} = 0.125, \quad \frac{1}{10} = 0.1, \quad \frac{1}{12} \approx 0.0833, \quad \frac{1}{14} \approx 0.0714, \quad \frac{1}{16} = 0.0625$ Adding them up: $0.125 + 0.1 + 0.0833 + 0.0714 + 0.0625 \approx 0.4422$ Calculating HM: $\text{HM} = \frac{5}{0.4422} \approx 11.309$

4. Verify and Summarize

Now we have the following values:

• AM = 12
• GM $\approx 11.458$
• HM $\approx 11.309$

Now we verify the inequalities:

• $\text{AM} \geq \text{GM}$: $12 \geq 11.458$ (True)
• $\text{GM} \geq \text{HM}$: $11.458 \geq 11.309$ (True)

Final Answer

Hence, we have proved that: $\text{AM} \geq \text{GM} \geq \text{HM}$