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Mathematics Worksheet, Cheat Sheet of Mathematics

College of MarinMathematics

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

Typology: Cheat Sheet

2020/2021

Uploaded on 11/06/2021

emjay01010
emjay01010 🇵🇭

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Worksheet in GE Math:
Wk-4
Nature of Mathematics
1.2 The Fibonacci sequence
Content:
The human mind is hardwired to recognize patterns. In mathematics
patterns may be generated by performing one or several mathematical
operations repeatedly.
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.
Example :
Analyze the given sequence for its rule and identify the next three
terms.
a. 1,10, 100, 1000
b. 2, 5, 9, 14, 20
Solutions:
a. Looking at the set of numbers, it can be observed that each
term is a power of 10:
100 = 1 following the rule the next three terms are: 104 = 10,000,
101 = 10 105 = 100,000, and 106 =1,000,000.
102 = 100
103 = 1,000
b. looking at the sequence 1st term (2+3=5),followed by (5+4=9) then
(9+5=14), (14+6=20); therefore the next three terms are (20+7=27),(27+8=35)
and (35+9=44)
Objectives:
1. Identify Fibonacci sequence
2. Solve problems involving Fibonacci
pf3
pf4
pf5

Partial preview of the text

Download Mathematics Worksheet and more Cheat Sheet Mathematics in PDF only on Docsity!

Worksheet in GE Math:

Wk-

Nature of Mathematics

1.2 The Fibonacci sequence

Content:

The human mind is hardwired to recognize patterns. In mathematics

patterns may be generated by performing one or several mathematical

operations repeatedly.

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.

Example :

Analyze the given sequence for its rule and identify the next three

terms.

a. 1,10, 100, 1000

b. 2, 5, 9, 14, 20

Solutions:

a. Looking at the set of numbers, it can be observed that each

term is a power of 10:

0

= 1 following the rule the next three terms are: 10

4

1

5

= 100,000, and 10

6

2

3

b. looking at the sequence 1st term (2+3=5),followed by (5+4=9) then

(9+5=14), (14+6=20); therefore the next three terms are (20+7=27),(27+8=35)

and (35+9=44)

Objectives:

**1. Identify Fibonacci sequence

  1. Solve problems involving Fibonacci**

Check your progress: Analyze the given sequence for its rule and identify the next three terms. a. 16, 32, 64, 128 b. 1, 1, 2, 3, 5, 8 The sequence in letter b is a special sequence called the Fibonacci sequence. Fibonacci sequence – name after the Italian Mathematician LEONARDO OF PISA Leonardo of Pisa –was better known by his nickname FIBONACCI.

  • He discovered the special sequence Fibonacci as he looked at how a hypothesized group of rabbits bred and reproduced. He noted that the set of numbers generated from this problem could be extended by getting the sum of the two previous terms. 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 3 + 5 = 8 0, 1, 1, 2, 3, 5, 8 5 + 8 = 13 0, 1, 1, 2, 3, 5, 8, 13 ; 0, 1, 1, 2, 3, 5, 8, 13……. While the sequence is widely known as Fibonacci sequence, this pattern is said to have been discovered much earlier in India. According to some scholarly articles, Fibonacci sequence is evident in the number of variation of a particular category of Sanskrit and Prakrit poetry meters. In poetry, meter refers to the rhythmic pattern of syllables. Fibonacci sequence has many interesting properties. Among these patterns are visible in nature. Some of nature’s most beautiful patterns, like the spiral arrangement of sun flower seeds, the number of petals in a flower, and the shape of a snail’s shell all of them contain Fibonacci numbers. It is also interesting to note that the ratios of successive Fibonacci numbers approach the numbers Ø (phi), also known as the Golden Ratio. This is approximately equal to 1.618. 1/1 = 1.0000 13/8 = 1. 2/1 = 2.0000 21/13 = 1. 3/2 = 1.5000 34/21 = 1. 5/3 = 1.6667 55/34 = 1. 8/5 = 1.6000 89/55 = 1. As such, this ratio is visible in many works of art and architecture such as in the Mona Lisa, the Notre Dame Cathedral, and the Parthenon. In Fact the Human DNA molecule also contains Fibonacci numbers, being 34 Angstroms long by 21 Angstroms wide for each full cycle of the double helix spiral. As shown in the list above, this approximates the golden ratio at a value of about 1.619 (1 angstrom =10-10^ meter or 0.1 nanometer).

Check your progress: Let Fib (n) be the nth term of the Fibonacci sequence, with Fib (1) = 1, Fib (2)= 1, Fib (3) = 2, and so on..

  1. Find Fib (5) =
  2. Find Fib (10) =
  3. If Fib (12) = 144 and Fib (14) =377 what is Fib (13) = ANS:
    1. Fib (5) = 5
    2. Fib (10) = 55
    3. Fib (13) = 1.2 Exercises: I. Give the correct answer of the given question: ________1.The arrangement of these terms is set by a definite rule. ________2. Sequence is an ordered list of numbers called ____ that may have repeated values. ________3.As special type of sequence name after the Italian Mathematician ________4.The Italian Mathematician ________5. Leonardo of Pisa was better known by his nickname. ________6. He discovered the special sequence as he looked at how a hypothesized group of ______ bred and reproduced. ________7. The succeeding terms in the sequence can be generated by ____ the two numbers that came before the term. ________8. The patterns have been discovered much earlier in what place? ________9.Fibonacci sequence is evident in the number of variation of a particular category of____________ poetry meters. ________10.In poetry,_______ refers to the rhythmic pattern of syllables. ________11.Fibonacci sequence has many interesting properties. Among these patterns are visible in _______. ________12.Some of nature’s most beautiful patterns like the spiral arrangement of _____ seeds. ________13.It is also interesting to note that the ratios of successive Fibonacci numbers approach the numbers ___. ________14. Phi also known as the ________. This is approximately equal to 1.618. ________15. Golden ratio is visible in many works of art and architecture such as in the Mona Lisa, the Notre Dame Cathedral, and the _______.

II. Give the correct answer by following the instruction. Let Fib (n) be the nth term of the Fibonacci sequence, with Fib (1) = 1, Fib (2)= 1, Fib (3) = 2, and so on..

  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 ff sums: a. Fib (1) + Fib (2) = b. Fib (1) + Fib (2) + Fib (3) = c. Fib (1) + Fib (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)? Note: Answer completely
  6. Evaluate the product of the ff: a. Fib (3) x Fib (5) = b. Fib (4) + Fib (2) x Fib (1) = c. Fib (3) x Fib (5) + Fib (2) =
  7. Starting with the first Fibonacci number, Fib (1) = 1 and the second Fibonacci number, Fib(2) = 1, What is the 15th^ Fibonacci number, Fib (15) =?
  8. What is Fib (20)?
  9. Given Fib (30) = 832,040 and Fib (28) = 317,811 what is Fib (29) =?
  10. What is Fib (8)? Solve the given problem: Consider Fib (3) = 2. What do you notice about every 3rd^ Fibonacci number, i.e Fib (6), Fib (9), Fib (12), ……..? Similarly, look at Fib (4) = 3, then check out every 4th^ Fibonacci number i.e Fib (8), Fib (12), Fib (16) ….? What seem to be the pattern behind these sequences generated from Fibonacci numbers?