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Problem of the Week
Problem A and Solution
Number Sequence
Problem
Looking at the following number sequences, what could be the next number if the patterns
continue?
A) 1 2 4 8 16 32 ?
B) 1 1 2 3 5 8 13 21 ?
What is the rule for each pattern?
Solution
A) One possible way of obtaining the sequence is to double the previous number.
Alternatively, if you add all the previous numbers and add 1, you get the next
number. Using this pattern, the next number in the sequence could be 64.
B) One possible way of obtaining each number in this sequence is to add the
previous two numbers. Using this pattern, the next number in the sequence
could be 13 + 21 = 34.
There are other ways to describe the sequence patterns. For example, if we look
at the differences between the numbers in each case, we can see patterns there
too. For the first sequence, the differences double each time. In the second case,
the differences match the numbers in the sequence themselves.
When looking for patterns, we always want to keep an open mind. As long as it
can be justified, students may reasonably find something that would result in a
different next number.
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Teacher’s Notes
Using two of the ways to determine the next number in pattern A leads us to a
formula that determines the sum of the first n powers of 2 (noting that 20 = 1).
For any non-negative integer n:
20 + 21 + 22 + 23 + ... + 2n = 2(n+1) − 1.
For example 20 + 21 + 22 + 23 + 24 = 25 − 1 = 31.
This is an example of a geometric series. We see geometric series in many
applications including fractals and compound interest.
The pattern in part B is known as the Fibonacci Sequence. These numbers
appear in many places in nature including patterns of spirals that we see in
flowers and pinecones.