define calculation for the numbers ( a ; b; c) such that the new triplet is as follows.
(a ; b ; c) --> transforms into --> (b+c ; c+a ; a+b)
the first terms for a,b,c are : (1 ; 3 ; 5)
Perform the calculation 2015 times, such that the new resulting numbers of the triplet are (x ; y ; z)
calculate the subtraction for numbers x-z
The Attempt at a Solution
I would probably not have attempted this problem. This was an old college entrance exam question. But it was supposedly solvable with high school and trade school knowledge.
Our class at school back in the day only had recursive sequences post calculus in high school mathematics.
As I recall I did rather mediocre or less than average at that class.
I was also wondering what would be the "smart way" to solve this problem assuming one only had pen and paper, and perhaps function calculator at hand. (no computer, no graphing calculator with recursive programs, no excel...).
I tend to do these kind of recursions first by making a table of the numbers and transformations and then try to find some kind of connections.
Off the top of my head, it looks like the B value (b=c+a)
It looks like B value grows as follows.
where w= number of transformations,
also the beginning number 3 could be regarded as the zeroeth transformation such that
It looks like C grows somewhat as follows.
C2=3*2^1+ 3*2^0+ 5
C3=3*2^2 + 3*2^1 + 3*2^0 +1
C4=3*2^3 + 3*2^2 + 3*2^1 + 3*2^0 +5
C5=3*2^4 + 3*2^3 + 3*2^2 + 3*2^1 + 3*2^0 +1
In a similar fashion some sort of pattern can be found for the A value growth.
I was simply wondering what kind of subtraction there would become for the 2015-th term.
The B value alone would be staggeringly high. The B value should be according to my estimation something like 3*2^2015
What would be the "smart way" to solve what would be the 2015-th term for the X value?
You could say X value, or A value, either way it is the first number in the triplet.
The question itself asked for the
X2015 - Y2015
Now that I looked again at the growth of the values of A and C... It looks like the B is always like... the arithmetic mean, of the A and C values. Whether this is useful or not... it does not seem obvious to me.
This arithmetic mean connection holds for all the first 10 terms, whichI did on pen and paper. Hehe... it was laborious but the connection jumped up at me when I looked at the paper.
Thus it seems likely that the
[X2015+Z2015 ] / 2=Y2015
I'm not very good at these recursions it seems. It was one troublesome problem in my math book.
Obviously had I known perfect and good way to solve the problem easily, I would not have posted a thread about it. I could definitely use some tips or pointers.