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Problem 50 Hard Difficulty

Given the following sequence of numbers*
$$1, a, a^{2}, a^{3}, \ldots, a^{n}$$
*Permission to reprint SAT materials does not constitute review or endorsement by Educational Testing Service or the College Board of this publication as a whole or of any other questions or testing information it may contain. This problem appeared, minus the additional assumption, on an SAT in 1996. Colin Rizzio, a high school student at the time, became an instant celebrity when he noticed that the additional assumption was needed to complete the problem. Source: The New York Times. where $n$ is a positive even integer, with the additional assumption that $a$ is a positive number, the median is best described as
a. greater than $a^{n / 2}$
b. smaller than $a^{n / 2}$
c. equal to $a^{n / 2}$
d. The relationship cannot be determined from the information given.




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Video Transcript

this question proposes a sequence that goes one Hey, a squared, a cubed all the way up to eight to the end and we're told that n is an even number. So since N is even, what does that mean? Well, that means that we're gonna have a total odd number in our sequence, right? Look at this. When a is in, is to we have uneven number for end and we have three numbers in our sequences. Three numbers are sequence. If End was four, then we'd have five numbers in our sequence, so even in gives us on odd number in our sequence. And now, when we're finding the median of an odd numbered sequence, we know that we take the total number which we normally call end. But in this time, I'll call it M. We take the total number, add one and divide by two. And that is our, uh, our sequence total. Right, So in. But in this instance, we have, um, how many terms we're gonna have. What would end is to we had three and when and was four. We had five. So when S O. M. Is gonna equal n plus one So if we have end in the median, Big M is going to be a X and plus one plus one over two, which is an over to plus one so X and over two plus one more second look like well, we know and over two of the whole number because it because it is because it is even. So, what's X and over to post one Will x one They're X one is one or another way of saying that is a 20 x two is a to the first. An X three is a to the second so we can see that X So X m is just a to the M minus one. That's a general rule. Now, if we have a feeling for X and over to plus one, we're looking for X n over two plus one. We know that that's gonna be a to the end over to plus one minus one, which is just equal to the median is gonna be He's gonna be equal to a o. R race the end over to And so that is what the media is gonna be. And I think that corresponds to answer C in this multiple choice question, Sylvester finally