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| 15 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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.
As Emily, Kevin, and Izabella continued their search, the books revealed more clues leading them closer to the magical library. They noticed that the mysterious numbers left by previous adventurers seemed to follow distinct patterns.
Find the fifth term of each sequence, then use the first five terms to graph each sequence on a coordinate plane.
Graph:
Graph:
In this sequence, the difference between consecutive terms is 7, so it is an arithmetic sequence with a common difference of 7. This means that the fifth term of the sequence is calculated by adding 7 to the fourth term.
To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position n and the vertical axis represents the term an. Then, plot the ordered pairs (n,an) on the coordinate plane.
In this sequence, the difference between the first and second terms is 6, but the difference between the second and third terms is 18. There is no common difference between consecutive terms. However, there is a common ratio between consecutive terms — 3 — so this is a geometric sequence. The fifth term of this sequence will be 3 times the fourth term.
To graph this sequence, start by drawing a coordinate plane where the horizontal axis represents the position n and the vertical axis represents the term an. Then, plot the ordered pairs (n,an) on the coordinate plane.
Looking over the completed graphs, Izabella said, The arithmetic sequence looks like a straight line, but the geometric sequence looks like a curve. I wonder if this will be important.
Every arithmetic sequence can be described by a linear function that is defined for the set of counting numbers. This function, referred to as the explicit rule of an arithmetic sequence, follows a specific general format.
an=a1+(n−1)d
n | an | Rewrite |
---|---|---|
1 | a1 | a1+0⋅d |
2 | a2 | a1+1⋅d |
3 | a3 | a1+2⋅d |
4 | a4 | a1+3⋅d |
5 | a5 | a1+4⋅d |
The coefficient of the common difference is always 1 less than the value of the position n. This makes it possible to write an explicit rule like the following formula.
an=a1+(n−1)d
In the heart of their town, the three friends finally reached the entrance of the mystical library.
All the mysteries lay behind a door with symbols on it. To unlock the door, they had to solve one more puzzle. Help them open the door.
n=8
Subtract term
Multiply
Add terms
The following applet shows the first five terms of an infinite arithmetic sequence. Determine the explicit rule of the sequence to calculate the indicated term of the sequence.
While exploring the magical library, Emily, Kevin, and Izabella came across a section dedicated to the Fibonacci sequence. This sequence, discovered by the Italian mathematician Leonardo Fibonacci, has occupied minds for centuries with its fascinating properties.
Heichi keeps track of the number of pages he reads in a certain amount of time. The hours spent reading and the amount of pages he reads are shown in the table.
Hours, n | Number of Pages, an |
---|---|
1 | 20 |
2 | 35 |
3 | 50 |
We are given a table showing the number of hours spends reading and the corresponding number of pages Heichi reads.
Hours, n | Number of Pages, a_n |
---|---|
1 | 20 |
2 | 35 |
3 | 50 |
We want to write an explicit rule to find the number of pages Heichi reads in n hours. Let's consider just the number of pages for now. They can form an increasing sequence. We can check to see if there is a constant difference between consecutive terms.
This is an arithmetic sequence with first term 20 and common difference 15. Let's use the general formula for the explicit rule of an arithmetic sequence. a_n= a_1+(n-1) d In this rule, a_1 is the first term in a given sequence, d is the common difference from one term to the next, and a_n is the nth term in the sequence. Let's substitute a_1= 20 and d= 15 into the rule and simplify.
This rule gives us the number of pages Heichi can read in n hours.
Ignacio records the distance he runs during various time intervals. The times spent running and the corresponding distance in meters are shown in the table.
Time in minutes, n | Distance in meters, an |
---|---|
3 | 400 |
5 | 640 |
7 | 880 |
We are given a table showing the times Ignacio spent running and the corresponding distance he ran. We want to write a function to express their relationship.
Time in minutes, n | Distance in meters, a_n |
---|---|
3 | 400 |
5 | 640 |
7 | 880 |
In this table, the values in the time column increase by 2 while the values in the distance row increase by 240. If we were to think of the distances Ignacio runs as a sequence, the third, fifth, and seventh terms would be 400, 640, and 880, respectively. This means that the difference between consecutive terms would be 240÷ 2 =120, and the first term would be 400-2(120)=140.
This is an arithmetic sequence with first term 160 and common difference 120. Now we can use the general formula for the explicit rule of an arithmetic sequence. a_n= a_1+(n-1) d Let's substitute a_1=160 and d=120 into the formula and simplify.
This rule provides the distance, in meters, that Ignacio ran in n minutes.
Describe the value of each term of the sequence as a function of its position.
Position | 3 | 4 | 5 | 6 | n |
---|---|---|---|---|---|
Term | 31 | 91 | 271 | 811 |
We want to describe the value of each term as a function of its position. To achieve this, we will attempt to rewrite each given term as a numeric expression that includes its position. Let's take a look at the values.
Position | 3 | 4 | 5 | 6 | n |
---|---|---|---|---|---|
Term | 1/3 | 1/9 | 1/27 | 1/81 |
Notice that each term is found by multiplying the previous term by 13 and can be expressed as a power of 13.
Position | 3 | 4 | 5 | 6 |
---|---|---|---|---|
Term | 1/3 | 1/9 | 1/27 | 1/81 |
Rewrite | 1/3 = (1/3)^1 | 1/3^2= (1/3)^2 | 1/3^3= (1/3)^3 | 1/3^4= (1/3)^4 |
As we can see, the exponents increase by one for each consecutive term. Furthermore, each exponent is 2 less than the position of that term. For example, the exponent of the third term is 1, which is the result of subtracting 2 from its position 3.
Position | 3 | 4 | 5 | 6 |
---|---|---|---|---|
Term | 1/3 | 1/9 | 1/27 | 1/81 |
Rewrite Exponents | (1/3)^1 = (1/3)^(3- 2) | (1/3)^2 = (1/3)^(4- 2) | (1/3)^3 = (1/3)^(5- 2) | (1/3)^4 = (1/3)^(6- 2) |
Considering this pattern, the value of the term in position n can be described as ( 13 )^(n- 2). a_n = ( 1/3)^(n- 2) We can substitute any natural number n to get the corresponding value of term, so this is a function of the sequence.
Let's use the function we found in Part A to find the first term of the sequence. In this formula, we will replace n with 1 and then simplify.
The first term of the sequence is 3.