AIME I #8 2003

[stupidityismygam]

In an increasing sequence of four positive integers, the first three terms form an arithmetic progression, the last three terms form a geometric progression, and the first and fourth terms differ by [unparseable or potentially dangerous latex formula] Find the sum of the four terms.

[bradp]

i smell some guessing and checking

[stupidityismygam]

i smell a dumb idea unless you can get it down to like 2 values for the first term (that's what i did, then i guessed and checked those :) )

[Roel]

Lets call the first term of our sequence [unparseable or potentially dangerous latex formula], and our common difference [unparseable or potentially dangerous latex formula]. That means our first 3 terms are [unparseable or potentially dangerous latex formula]. We can see that the common ration between our second and third terms is [unparseable or potentially dangerous latex formula], which makes our 4 terms [unparseable or potentially dangerous latex formula].

They give us...
[unparseable or potentially dangerous latex formula]
I had written my solution already before i clicked something and it erased, so i'm not going to write down the process of how to solve for [unparseable or potentially dangerous latex formula] but this is what it is:
[unparseable or potentially dangerous latex formula]
In order for [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] to be positive integers [unparseable or potentially dangerous latex formula] has to be either [unparseable or potentially dangerous latex formula]. From our equation we can see that we need a multiple of [unparseable or potentially dangerous latex formula] therefore we use the [unparseable or potentially dangerous latex formula].

Plugging in:
[unparseable or potentially dangerous latex formula], and similarly for the rest of the terms, which come out to be [unparseable or potentially dangerous latex formula].
[unparseable or potentially dangerous latex formula].