As tweeted on twitter.com/dmarain...
J noticed that for an arithmetic sequence like 3,7,11,15,19 the median equals the arithmetic mean. In this case, the median and "mean" are both 11. She found this was well-known and not too difficult to prove.
She wondered if there was an analogous rule for geometric sequences like 2,4,8,16,32. Instead of the arithmetic mean she tried the geometric mean:
(2•4•8•16•32)^(1/5) which equals 8, the median. VERIFY THIS WITHOUT A CALCULATOR!
Unfortunately her conjecture failed for a geometric sequence with an even number of terms like 2,4,8,16 in which the median equals 6 while the GM = 4√2.
(a) Test her conjectures with at least 4 other finite geometric sequences, some with an odd number of terms, some with an even #.
(b) PROVE her conjecture for an odd number of terms.
Hint: If n is odd then a,ar,ar²,...,ar^(n-1) would have an odd # of terms. Why?
(c) How would the definition of median have to be modified for an even # of terms?
How much arithmetic/algebraic background is needed here?
Arithmetic sequences more than enough for middle schoolers to explore? Geometric too ambitious?
PROOF too sophisticated for middle schoolers? How would you adapt it? We are trying to raise the bar, right?