A Sequence in Which Each Number Is the Sum of the Previous Two (Fibonacci Sequence)
A sequence in which “the sum of the previous two numbers becomes the next number” is called the Fibonacci sequence.
It is a fascinating sequence of numbers that often appears in nature, such as in the number of petals on flowers and the spirals of seashells. In entrance examinations for junior high schools, it often appears in problems such as “ways to climb stairs.”
What You Will Learn
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
A number = the previous number + the number two places before it
A typical problem that uses this rule is “ways to climb stairs.” When you can climb “one step at a time or two steps at a time,” the number of ways to climb a staircase with n steps is the sum of the previous two numbers.
Example Problem
There is a staircase with 7 steps. You can climb either 1 step or 2 steps at a time. How many ways are there to climb the staircase in total?
Solution:
① There is 1 way to reach the first step and 2 ways to reach the second step.
② From the third step onward, calculate the number of ways by “the number of ways to reach the previous step + the number of ways to reach the step two places before.”
Third step: 1 ＋ 2 ＝ 3 ways
Fourth step: 2 ＋ 3 ＝ 5 ways
Fifth step: 3 ＋ 5 ＝ 8 ways
Sixth step: 5 ＋ 8 ＝ 13 ways
Seventh step: 8 ＋ 13 ＝ 21 ways
There is a staircase with 7 steps. You can climb either 1 step or 2 steps at a time. How many ways are there to climb the staircase in total?
Solution:
① There is 1 way to reach the first step and 2 ways to reach the second step.
② From the third step onward, calculate the number of ways by “the number of ways to reach the previous step + the number of ways to reach the step two places before.”
Third step: 1 ＋ 2 ＝ 3 ways
Fourth step: 2 ＋ 3 ＝ 5 ways
Fifth step: 3 ＋ 5 ＝ 8 ways
Sixth step: 5 ＋ 8 ＝ 13 ways
Seventh step: 8 ＋ 13 ＝ 21 ways
Key Points
Remember
- It is important to notice the rule: “Adding the previous two numbers gives the next number.”
- Making a table and calculating in order is the least likely way to make a mistake.
- It is useful to remember the beginning of the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…).
Practice Problems
- There is a sequence whose first number is 2 and whose second number is 3. Each subsequent number is the sum of the previous two numbers. Find the eighth number.
- There is a staircase with 10 steps that you can climb either 1 step or 2 steps at a time. How many ways are there to climb it in total?
- Find the sum of the first through the tenth terms of the sequence 1, 1, 2, 3, 5, 8, …
Answers and Explanations
- Answer: Eighth term = 81
Explanation: First term = 2, second term = 3, and each subsequent term is the sum of the previous two terms. Third term = 5, fourth term = 8, fifth term = 13, sixth term = 21, seventh term = 34, eighth term = 34+47=… Calculating accurately: 2, 3, 5, 8, 13, 21, 34, 55. Eighth term = 55. - Answer: 89 ways
Explanation: 1 step = 1, 2 steps = 2, 3 steps = 3, 4 steps = 5, 5 steps = 8, 6 steps = 13, 7 steps = 21, 8 steps = 34, 9 steps = 55, 10 steps = 89. The number of ways to climb each step forms a Fibonacci-like sequence in which each term is the sum of the previous two terms. - Answer: Sum from the first term through the tenth term = 143
Explanation: The terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Their sum is 1+1+2+3+5+8+13+21+34+55 = 143. (Property of the Fibonacci sequence: the sum from the first term through the nth term = the (n+2)th term − 1. Since the twelfth term is 144, we can confirm that 144-1=143.)