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If first and (2n-1)^th terms of an AP, ...

If first and `(2n-1)^th` terms of an AP, GP. and HP. are equal and their nth terms are a, b, c respectively, then

A

`a = b = c`

B

`a ge b ge c`

C

`a + c = b`

D

`b^(2) = ac`

Text Solution

Verified by Experts

The correct Answer is:
A, B, D
Since, first and `(2n -1)th` terms are equal.
Let first term be x and `(2n -1)`th term be y shoe middle term is `l_(n)`.
Thus, in arithmetic progression, `t_(n) = (x + y)/(2) = a`
In geometric progression, `t_(n) = sqrt(xy) = b`
In harmonic progression, `t_(n) = (2xy)/(x + y) = c`
`rArr b^(2) = ac and a gt b gt c` [using `AM gt GM gt HM`]
Here, equality holds (i.e., `a = b =c`) only if all terms are same. Hence, options (a), (b) and (d) are correct
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