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Question 1 aThe Fibonacci sequence can be achieved from Pascals triangle by adding up the diagonal rows Refer to Figure 11 Figure 11 This is possible as like the Fibonacci sequence Pascals triangle adds the two previous numbers above to get the next number the formula 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 reason that it is added diagonally is because of how the numbers are added down and not across like in the Fibonacci sequence but it is a lot like the Fibonacci sequence so it makes you think if the Fibonacci sequence was written out differently if it would have all these pattern in it but its not part of the assignment to investigate that It is possible to assume 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 see in the Appendix The sum of the row is equal to the powers of 2 for example Powers of 2 Pascals Triangle 201 212 224 238 2416 Row 11 Row 22 Row 34 Row 48 Row 516 This is amazing as it is saying that the sum of each row in Pascals triangle is a square number ii The relationship between Pascals Triangle and the powers of 11 was a harder relationship to find
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