Question

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+c=b (c) a>b>c and `ac-b^2=0` (d) none of these
A. `a = b = c`
B. `a ge b ge c`
C. `a + c = b`
D. `ac - b^(2) = 0`

Answer 1

Correct Answer - 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