Fibonacci Sequence

Show your work for each problem.

1.
Which number comes next? $1, 1, 2, 3, 5, 8, 13, \ldots$
1. a)20
2. b)18
3. c)21
4. d)26
Answer: 21
The next term is $8 + 13 = 21$ .
2.
With $F_1 = F_2 = 1$ , what is $F_{10}$ , the 10th Fibonacci number?
Answer: 55
$F_8 = 21$ and $F_9 = 34$ , so $F_{10} = 21 + 34 = 55$ .
3.
Which rule produces the Fibonacci sequence?
1. a)Add the two previous terms
2. b)Square the previous term
3. c)Multiply the previous term by 2
4. d)Add 2 to the previous term
Answer: Add the two previous terms
Each term is the sum of the two terms before it: $F_n = F_{n-1} + F_{n-2}$ . Doubling would give $1, 2, 4, 8, \ldots$ instead.
4.
Two neighbouring Fibonacci numbers are $34$ and $55$ . What is the next term?
Answer: 89
$34 + 55 = 89$ .
5.
In Fibonacci's rabbit problem the numbers of pairs in months 1 to 4 are $1, 1, 2, 3$ . How many pairs are there in month 7?
Answer: 13
1. How many pairs are there in month 5? 5
2. How many pairs are there in month 6? 8
3. How many pairs are there in month 7? 13
6.
Start from $F_4 = 3$ and $F_5 = 5$ and find $F_8$ .
Answer: 21
1. What is $F_6$ ? 8
2. What is $F_7$ ? 13
3. What is $F_8$ ? 21
7.
Which ratio of neighbouring Fibonacci numbers is closest to the golden ratio $\varphi \approx 1.618$ ?
1. a)13/8
2. b)89/55
3. c)3/2
4. d)8/5
Answer: 89/55
$\frac{89}{55} \approx 1.6182$ , while $\frac{8}{5} = 1.6$ , $\frac{3}{2} = 1.5$ and $\frac{13}{8} = 1.625$ . The later the terms, the closer to $\varphi$ .
8.
Find the sum of the first 10 Fibonacci numbers: $1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55$ .
Answer: 143
The sum is $F_{12} - 1 = 144 - 1 = 143$ .
9.
Two neighbouring Fibonacci numbers are $233$ and $377$ . Work backwards: what is the term two places before $233$ ?
Answer: 89
1. Which term comes just before $233$ ? 144
2. Which term comes before that one? 89
3. Check: what is $89 + 144$ ? 233
10.
Every third Fibonacci number is even. Which of these terms is even?
1. a) $F_{24}$
2. b) $F_{20}$
3. c) $F_{22}$
4. d) $F_{25}$
Answer: $F_{24}$
$24 = 3 \times 8$ is a multiple of 3, so $F_{24}$ is even (it is $46368$ ). 20, 22 and 25 are not multiples of 3.
11.
The Lucas numbers use the Fibonacci rule but start with 2 and 1: $2, 1, 3, 4, 7, 11, 18, \ldots$ What is the next term?
Answer: 29
$11 + 18 = 29$ .
12.
The Fibonacci numbers $8$ and $13$ are neighbours. Find the next term, then the ratio of that term to $13$ , rounded to 3 decimal places.
Answer: 1.615
1. What is the next term after $13$ ? 21
2. What is $\frac{21}{13}$ rounded to 3 decimal places? 1.615
3. How far is $1.615$ from $1.618$ ? 0.003
13.
Use the sum rule $F_1 + \cdots + F_n = F_{n+2} - 1$ to find $1 + 1 + 2 + 3 + 5 + 8 + 13$ .
Answer: 33
1. How many terms are added? (This is $n$ .) 7
2. The rule needs $F_{n+2}$ . What is $n + 2$ ? 9
3. What is $F_9$ ? 34
4. What is the sum $F_9 - 1$ ? 33
14.
The golden ratio has the property $\varphi^2 = \varphi + 1$ . Using $\varphi \approx 1.618$ , what is $\varphi^2$ ?
1. a)2.618
2. b)3.236
3. c)2.236
4. d)1.309
Answer: 2.618
$\varphi^2 = \varphi + 1 \approx 1.618 + 1 = 2.618$ . Check: $1.618 \times 1.618 \approx 2.618$ . ( $3.236$ is $2\varphi$ , not $\varphi^2$ .)
15.
You climb a staircase taking 1 or 2 stairs at a time. There is 1 way for 1 stair and 2 ways for 2 stairs, and each next count is the sum of the two before. In how many ways can you climb 10 stairs?
Answer: 89
For 1 to 10 stairs: $1, 2, 3, 5, 8, 13, 21, 34, 55, 89$ . So 10 stairs can be climbed in $89$ ways (this is $F_{11}$ ).
16.
A new sequence uses the Fibonacci rule but starts with $4$ and $10$ . Find terms 3 to 6, then the ratio of term 6 to term 5, rounded to 2 decimal places.
Answer: 1.63
1. What is term 3? 14
2. What is term 4? 24
3. What is term 5? 38
4. What is term 6? 62
5. What is $\frac{62}{38}$ rounded to 2 decimal places? 1.63
17.
How many of the first 30 Fibonacci numbers $F_1, F_2, \ldots, F_{30}$ are even?
Answer: 10
1. At which position is the first even Fibonacci number? 3
2. At which position is the next even one? 6
3. Even terms sit at multiples of 3. How many multiples of 3 are there from 1 to 30? 10
18.
Check numerically that $\varphi = \frac{1 + \sqrt{5}}{2}$ satisfies $\varphi = 1 + \frac{1}{\varphi}$ . Round every result to 3 decimal places.
Answer: 1.618
1. What is $\sqrt{5}$ rounded to 3 decimal places? 2.236
2. What is $\frac{1 + 2.236}{2}$ ? 1.618
3. What is $\frac{1}{1.618}$ rounded to 3 decimal places? 0.618
4. What is $1 + 0.618$ ? 1.618