Mind Your Decisions, I'm Presh Talwalkar.
A sequence starts with the number 1, and each subsequent term is multiplied by the common ratio a,
until the final term of a to the power of n, for n a positive even integer.
We're going to define two variables.
x is equal to the median of the sequence, and y is equal to a to the power of n over 2.
Which of the following statements is always true?
x is greater than y, x is less than y, x is equal to y, or none of the above?
This problem is adapted from the 1996 SAT exam.
Pause the video if you'd like to give this problem a try, and when you're ready, keep watching to learn how to solve this problem.
We can start analyzing this problem by considering specific values.
Suppose n is equal to 4. In
that case, y is equal to a to the power of 2.
What is x equal to?
Let's consider a specific value like a is equal to 3. We substitute that in; calculate that y is equal to 9.
Then, we calculate the median. The median will be the middle value of the sequence which is equal to 9.
That's exactly in the middle. Therefore x is equal to 9 and x is exactly equal to y.
Let's do another example. Let's again consider n is equal to 4 so that y is equal to a squared. Now
let's consider a is equal to 1/2.
When we substitute in we get a decreasing sequence. To calculate the median, we'll put it in ascending order.
Here we have the median is exactly equal to the middle value, which is 1 over 4, and that's exactly equal to y.
Therefore again, x is equal to y.
From these particular examples, one might conclude that x is always equal to y.
Well, this is actually true,
if a is positive. In that case,
the sequence is either increasing or decreasing, and the median will be a to the power of n over 2,
which is exactly equal to y. Therefore x is always equal to y if a is greater than 0.
The official answer to this problem did say that x is equal to y.
However, it's an incomplete answer. It is not the correct answer. A
17-year old Colin Rizzio figured out the correct answer should be none of the above,
and it made big waves across the media at the time.
Let's see why none of the above is the correct answer.
We'll again consider that n is equal to 4, so we get y is equal to a squared.
Now we consider a negative value for a.
Let's say a is equal to negative 3.
If we substitute that in, we get an alternating sequence.
y is equal to 9. What's the median value? What's the middle value of this sequence? To
calculate the median, we'll arrange the sequence in ascending order.
We can now see the middle value is equal to 1 and x is equal to 1. Therefore x is less than y.
So we'll put this particular case together with the other case that x was equal to y for a positive value of a.
Putting these observations together we can solve the question. Which of the following statements is always true?
Well, can we say that x is always greater than y? No
we can't because sometimes it's less and y and sometimes it's equal to y.
Again, we can't say that x is always less than y because sometimes it's equal to y, and we can't say x is always equal
to y because sometimes it's less than y. Therefore all of these are false, and the correct answer is none of the above.
The moral of the story is you should not blindly accept the official answer.
It's best to do your own analysis and Mind Your Decision.
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