Assignment regarding the fibonacci sequence and the golden ratio - maths assignemnt

Question 1

(a)         The Fibonacci sequence can be achieved from Pascals triangle by adding up the diagonal rows. Refer to Figure 1.1

Figure 1.1

This is practical as standardised the Fibonacci sequence, Pascals triangle adds the two previous ( reduces above) to get the next number, the manifestation if Fn = Fn-1 + Fn-2. Pascals Triangle is achieved by adding the two numbers above it, so uses the same basic principle. This is why there is a relationship. The mind that it is added diagonally is because of how the numbers are added down and not across equivalent in the Fibonacci sequence, precisely it is a lot like the Fibonacci sequence so it makes you think if the Fibonacci sequence was written let on differently if it would have all these pattern in it, but its not part of the assignment to investigate that. It is possible to encounter that it is possible for the Fibonacci sequence to have been created from Pascals triangle as I dont know where the notion of the Fibonacci sequence was created for but it appears that other number patterns have been created from Pascals triangle so why couldnt it be possible that it was. Of course it works the opposite diagonal way as well.

(b) i. Powers of 2 has a relationship to Pascals triangle, See Appendix 1 at end of assignment for picture. As you can know in the Appendix The sum of the row is equal to the powers of 2.



for example

Powers of 2                                              Pascals Triangle

2^0         1

2^1         2

2^2         4

2^3         8

2^4         16

Row 1                  1

Row 2                  2

Row 3                  4

Row 4                  8

Row 5                  16

This is amazing as it is saying that the sum of...

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