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If three positive unequal quantities a,b...

If three positive unequal quantities `a,b,c` be in HP, then prove that `a^(n)+c^(n)gt2b^(n),n in N`.

A

Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.

B

Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.

C

Statement -1 is true, Statement -2 is False.

D

Statement -1 is False, Statement -2 is True.

Text Solution

Verified by Experts

The correct Answer is:
A
If a,b,c are in H.P., then b is the HM of a and c.
But, the GM of a and c is `sqrt(ac)`. . . .(i)
Using `AMgtGM`, we have
`(a^(n)+c^(n))/(2)gtsqrt(a^(n)c^(n))`
`rArr" "(a^(n)+c^(n))/(2)gt(ac)^(n//2)`
`rArr" "(a^(n)+c^(n))/(2)gtb^(n)`
`rArr" "a^(n)+c^(n)gt2b^(n)`
Hence, statement -1 and 2 are true statement -2 is a correct explanation for statement -1.
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