THE FIBONACCI SEQUENCE As we have seen in the previous section, the human mind is hardwired to recognize patterns. In mathematics, we can generate patterns by performing one or several mathematical operations repeatedly. Suppose we choose number 3 as the first number in our pattern. We then choose to add 5 to our first number, resulting in 8, which is our second number. Repeating this process, we obtain 13,18,23,28… as the succeeding numbers that form our pattern. In mathematics, we call these ordered lists of numbers a SEQUENCE. A SEQUENCE is an ordered list of numbers called ‘terms’, that may have repeated values. The arrangement of these terms is set by a definite rule. Generating a Sequence:(Practice drill) Analyze the given sequence for its rule and identify the next three terms. a. 1,10,100,1000,___ b. 2,5,9,20,___ What do you think would be the solution? Elaborate. a. 16, 32, 64, 128, ___ b. 1,1,2,3,5,8,___ -Starting with 0 and 1, the succeeding terms in the sequence can be generated by adding the two numbers that came before the term: 0+1 = 1 0.1,1 1+1 = 2 0,1,1,2 1+2 = 3 0,1,1,2,3 2+3 = 5 0,1,1,2,3,5 -While the sequence is widely known as FIBONACCI sequence, this pattern is said to have been discovered much earlier much in India. According to some scholarly articles, Fibonacci sequence is evident in the number of variations of a particular category of Sanskrit and Prakrit poetry meters. In poetry, meter, refers to the rhythmic pattern of syllables. Drill 2 Exercise set: Let Fib(n) be the nth term of the Fibonacci sequence, with Fib(1)=1,Fib(2)=1,Fib(3)=2……… 1. Find Fib (8) 2. Find fib 19) 3. If Fib(22)=17,711 and Fib(24)= 46,368, what is Fib(23)? 4. Evaluate the following sums: a. Fib(1)+ Fib(2)= _____ b. Fib (1)+Fib(2)+Fib(3)=___ c. Fib (1)+(2)+Fib(3)+Fib(4) = _____ 5. Determine the pattern in the successive sums from the previous question. What will be the sum of Fib(1)+ Fib(2)+… Fib(10)? 6. Answer completely. If you have a wooden board that is 0.75 meters wide, how long should you cut it such that the Golden Ratio is observed? Use 1.618 as the value of the Golden Ratio.
Answers
Step-by-step explanation:
6...For a Rectangle, length/width= golden ratio width 0.75
Let's the length be x
x/0.75=1.618
x/0.75*1.618=(3/4)*1.618=3*0.4045 =1.2135.
Length of the rectangle is 1.2135metres.
Given:
Sequences i) 1,10,100,1000,___
ii) 2,5,9,_20,_
a) 16, 32, 64, 128, ___
b) 1,1,2,3,5,8,___
Fib(1)=1, Fib(2)=1, Fib(3)=2, Fib(22)=17,711, Fib(24)= 46,368
Length of wooden board is 0.75m and golden ratio = 1.618
To find:
1) Next two numbers for the given sequence series.
2) Fib(8), Fib(19), Fib(23)
3) a. Fib(1)+ Fib(2)= _____
b. Fib(1)+Fib(2)+Fib(3)=___
c. Fib(1)+Fib(2)+Fib(3)+Fib(4) = _____
4) sum of Fib(1)+ Fib(2)+… Fib(10)
5) How long should wooden board be cut such that golden ratio is observed.
Solution:
1) 1,10,100,1000,___ we can see that next number is multiplied by 10. So next two numbers are 10000, 100000.
2,5,9,_,20_ . Here the pattern is +3, +4, +5, +6. Hence next numbers are 20+7 and 27+8 i.e, 27, 35.
16, 32, 64, 128, ___. Here the pattern is x2. So next numbers are 128x2, 256x2 i.e, 256, 512.
1,1,2,3,5,8,___. This series is fibonacci series. So next numbers are 13, 21.
2) Fib(8) = Fib(7) + Fib(6) = 21
Fib(19) = Fib(18) + Fib(17) = 4181
Fib(24) = Fib(23) + Fib(22) ⇒ 46,368 = Fib(23) + 17,711 ⇒ Fib(23) = 28657
3) Fib(1) + Fib(2) = 1+1 = 2
Fib(1)+Fib(2)+Fib(3)= 1+1+2 = 4
Fib(1)+Fib(2)+Fib(3)+Fib(4) = 1+1+2+3 = 7
4) Fib(1)+ Fib(2)+… Fib(10) = 0+1+1+2+3+5+8+13+21+34 = 88
Pattern is for sum of n fibinocci series S(n) = F(n+2)-1
5)
⇒
⇒ x = 0.4635m
Therefore length of wooden board of width 0.75m to observe golden ratio is 0.4635m.
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