AIME II #9 2004

[stupidityismygam]

A sequence of positive integers with [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all [unparseable or potentially dangerous latex formula] the terms [unparseable or potentially dangerous latex formula] are in geometric progression, and the terms [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are in arithmetic progression. Let [unparseable or potentially dangerous latex formula] be the greatest term in this sequence that is less than [unparseable or potentially dangerous latex formula] Find [unparseable or potentially dangerous latex formula]

[stupidityismygam]

We will systematically go through the sequence putting all of the terms in terms of [unparseable or potentially dangerous latex formula]

So, let r be the multiplier in the geometric progression

Then, [unparseable or potentially dangerous latex formula]

Since the 2nd,3rd,4th terms are in an arithmetic the 3rd term is the mean of the 2nd and 4th (prove this yourself)

So, [unparseable or potentially dangerous latex formula]

Since the 3rd,4th,5th terms are in a geometric the 4th term is the geometric mean of the 3rd and 5th (prove this yourself

So, [unparseable or potentially dangerous latex formula]

Now we go through this rather repetitive and quite tedious process many more times

[unparseable or potentially dangerous latex formula] So [unparseable or potentially dangerous latex formula] (see if you notice a pattern after this step)

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula] So [unparseable or potentially dangerous latex formula] (gosh the pattern is just screaming, im too lazy to prove it so i will keep on my tedious nonsensical way)

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula] So [unparseable or potentially dangerous latex formula]

Whew...that was hard work...now we go to the easy part :)

so [unparseable or potentially dangerous latex formula]

Now...[unparseable or potentially dangerous latex formula]

once you find r you can a) continue the tedious process b) follow the pattern c) already have the proof for the pattern and use b) without 'cheating' d) some clever thing that im too lazy to find

ok so lets cheat because im lazy
every other term is a perfect square (the odds are perfect squares) in fact they are of the form [unparseable or potentially dangerous latex formula]

So, [unparseable or potentially dangerous latex formula]

Now all we have to do is find [unparseable or potentially dangerous latex formula]

it is the GM of 15th and 17th which is the same as 29*33=87*11=957

957+16=973 and we win