SAT 1997

Algebra Level pending

Let 1 1 , a a , a 2 a^2 , \cdots , a n a^n be a sequence, where n n is a positive even integer.

Let x = x = the median of the whole sequence and let y = a n 2 y = a^{\frac{n}{2}} , then which of the following is always true?

x = y x = y x < y x < y x > y x > y None of the above

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1 solution

Adhiraj Dutta
Apr 18, 2020

As a a is not specified, we have two cases -

When a > 0 a > 0 , then the series is either increasing (for a > 1 a > 1 ) or decreasing (for a < 1 a < 1 ), so x = a n 2 x = a^{\frac{n}{2}} and hence x = y x = y .

When a < 0 a < 0 however, the series alternates between positive and negative values, hence x > y x > y .

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