Expert: Josh - 12/19/2007

Question
QUESTION: Consider the sequence that begins 4,3,... and continues by the rule:
Every subsequent element of the sequence is the sum of the two preceding
elements.
A. Determine the next five(third through seventh)elements of this
sequence.
B. compute the ratio of the sixth and seventh elements of this
sequence.  Is is close to the Golden Ratio?



ANSWER: HI Sandi

Part A:
We need some notation to describe this sequence properly.

Let x[n] be the nth term. With x[0]=4 and x[1]=3, subsequent terms in the sequence are described by the recurrence equation

x[n]=x[n-1]+x[n-2], for integer n>1.

When n=2, x[2]=x[0]+x[1]=4+3=7
When n=3, x[3]=x[1]+x[2]=3+7=10
and so forth...I'll let you complete the rest.

Part B: Consider the ratio between the sixth and seventh term, viz., x[6]/x[5]. Check whether this is close to the golden ratio [1+sqrt(5)]/2 which has an approximate value of 1.618

Golden ratio is further explained at http://en.wikipedia.org/wiki/Golden_ratio

---------- FOLLOW-UP ----------

QUESTION: This is super hard for me to understand, can you follow it out for me, so I can do my next question that is almost the same
Thanks

Answer
I am not sure whether you find the idea difficult to understand, or the notations difficult to follow.

All I am saying is that the first two terms
x[0]=4
x[1]=3
The rest, x[2], x[3], x[4] etc. have to be determined using the two previous terms.

So, x[2]=x[0]+x[1], but the values of x[0] and x[1] are known. We are told that x[0]=4 and x[1]=3. Making a direct substitution for the values of x[0] and x[1], we arrive at x[2]=4+3=7.

Now, we know that
x[0]=4
x[1]=3
x[2]=7.
What is the next term x[3]?
Again, we add up the last two terms. x[3]=x[1]+x[2]=3+7=10.

Now, we know that
x[0]=4
x[1]=3
x[2]=7
x[3]=10

Repeating this a few times, I mean, you should definitely try this yourself, the answers for part A are:

Fifth term: x[4]=x[2]+x[3]=7+10=17
Sixth term: x[5]=x[3]+x[4]=10+17=27
Seventh term: x[5]=x[3]+x[4]=17+27=44.

Part B: All I can say is that the Golden Ratio has special significance, and was discovered by brilliant mathematics in ancient time. You can follow the wikipedia link and read up on the topic yourself. But for now, all you need to know is that from geometry, the Golden Ratio approximately equals 1.618.

Question asks you to calculate x[6]/x[5], the values of x[6] and x[5] were found in part A. Once you have the answer, you report how close it is to 1.618... i.e., whether computing the ratio x[n]/x[n-1] from this sequence gives a good approximation of the Golden ratio.

Is this okay?