Fibonacci Sequence

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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
2.
With $F_1 = F_2 = 1$ , what is $F_{10}$ , the 10th Fibonacci number?
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
4.
Two neighbouring Fibonacci numbers are $34$ and $55$ . What is the next term?
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?
6.
Start from $F_4 = 3$ and $F_5 = 5$ and find $F_8$ .
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
8.
Find the sum of the first 10 Fibonacci numbers: $1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55$ .
9.
Two neighbouring Fibonacci numbers are $233$ and $377$ . Work backwards: what is the term two places before $233$ ?
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) $F_{25}$
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?
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.
13.
Use the sum rule $F_1 + \cdots + F_n = F_{n+2} - 1$ to find $1 + 1 + 2 + 3 + 5 + 8 + 13$ .
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
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?
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.
17.
How many of the first 30 Fibonacci numbers $F_1, F_2, \ldots, F_{30}$ are even?
18.
Check numerically that $\varphi = \frac{1 + \sqrt{5}}{2}$ satisfies $\varphi = 1 + \frac{1}{\varphi}$ . Round every result to 3 decimal places.