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 \( A_{n-1}, G_{n-1} \) and \( H_{n-1} \) has arithmetic, geometric and harmonic means as \( A_{n}, G_{n}, H_{n} \), respectively.
[IIT-JEE 2007, 4+4+4M]
(i) Which one of the following statement is correct?
(a) \( G_{1}G_{2}G_{3}\ldots \)
(b) \( G_{1}G_{2}G_{3}\ldots \)
(c) \( G_{1}=G_{2}=G_{3}=\ldots \)
(d) \( G_{1}G_{3}G_{5}\ldots \) and \( G_{2}G_{4}G_{6}\ldots \)
(ii) Which of the following statement is correct?
(a) \( A_{1}A_{2}A_{3}\ldots \)
(b) \( A_{1}A_{2}A_{3}\ldots \)
(c) \( A_{1}A_{3}A_{5}\ldots \) and \( A_{2}A_{4}A_{6}\ldots \)
(d) \( A_{1}A_{3}A_{5}\ldots \) and \( A_{2}A_{4}A_{6}\ldots \)
(iii) Which of the following statement is correct?
(a) \( \mathrm{H}_{1}\mathrm{H}_{2}\mathrm{H}_{3}\ldots \)
(b) \( H_{1}H_{2}H_{3}\ldots \)
(c) \( \mathrm{H}_{1}\mathrm{H}_{3}\mathrm{H}_{5}\ldots \) and \( \mathrm{H}_{2}\mathrm{H}_{4}H_{6}\ldots \)
(d) \( H_{1}H_{3}H_{5}\ldots \) and \( H_{2}H_{4}H_{6}\ldots \)
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