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Here are a few recommended readings before getting started with this lesson.
A pattern describes a repeated change of numbers, shapes, colors, actions, or other elements. Patterns are based on a specific rule. The rule can also be used to find missing steps in the pattern. In the example below, matches have been placed together to create three figures.
Is it possible to find a pattern? Notice that each figure has one more triangle than the previous one. Therefore, the next figure should have 4 triangles.
There is also a pattern in the number of matches. In each step, the number increases by 2. The first figure has 3 matches, the second figure has 5 matches, the third figure has 7 matches, and so on.
The number of matches in the next couple of figures can be found using this pattern.
Emily and Kevin heard whispers of a mythical library filled with magical books in their town. Their research led them to some books full of mysterious numbers left behind by earlier explorers.
Sequences are ordered sets of numbers that follow identifiable patterns. Two types of sequences are arithmetic and geometric, each with their own unique characteristics. They play a key role in understanding the organized relationships between numbers.
Unlike the adding pattern in arithmetic sequences, geometric sequences multiply each term by a constant factor. The characteristics of geometric sequences will now be explored.
a1>0 | a1<0 | |
---|---|---|
r>1 | Increasing 3 →×2 6 →×2 12 →×2 24 →×2 48… |
Decreasing -3 →×2 -6 →×2 -12 →×2 -24 →×2 -48… |
r=1 | Constant
3 →×1 3 →×1 3 →×1 3 →×1 3… |
Constant
-3 →×1 -3 →×1 -3 →×1 -3 →×1 -3… |
0<r<1 | Decreasing 48 →×21 24 →×21 12 →×21 6 →×21 3… |
Increasing -48 →×21 -24 →×21 -12 →×21 -6 →×21 -3… |
r<0 | Alternating
3 →×(-2) -6 →×(-2) 12 →×(-2) -24 →×(-2) 48… |
Alternating
-3 →×(-2) 6 →×(-2) -12 →×(-2) 24 →×(-2) -48… |
After a while, Izabella joined Kevin and Emily on their adventure. To find the mysterious library, they decided to challenge themselves by reading the books as quickly as possible.
They each had their own reading goals. Emily aimed to increase her reading by 15 pages each day, Kevin planned to read double the pages of the previous day, and Izabella alternated between reading 15 and 25 pages every day.
Since there is a common difference between consecutive terms, the number of pages read forms an arithmetic sequence.
Sequence | Classification |
---|---|
15,30,45,60,… | Arithmetic Sequence |
In this case, there is no common difference or common ratio between consecutive terms, so the number of pages is neither arithmetic nor geometric.
Sequence | Classification |
---|---|
15,25,15,25,… | Neither arithmetic nor geometric |
Since there is a common ratio between consecutive terms, the number of pages read forms a geometric sequence.
Sequence | Classification |
---|---|
5,10,20,40,… | Geometric Sequence |
The following applet shows the first five terms of an infinite sequence. Analyze them carefully and determine whether the sequence is arithmetic, geometric, or neither.