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Describing Relationships and Extending Terms in Arithmetic Sequences.
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Here is an example of an arithmetic sequence.
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A sequence is any ordered list of numbers.
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An arithmetic sequence is a sequence found by adding the same number to the previous term.
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For example, you add nine to 28.
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And that equals 37.
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You add nine to 37.
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And that equals 46.
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Each of these numbers are terms in the sequence.
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Someone might ask you, find the next term in this arithmetic sequence.
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When they say, find the next term, we know that they’re looking for the value that comes after 64 here.
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And because we’re also given the information that it’s an arithmetic sequence, we also know that we will be adding to find the next value.
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Each step in this sequence has been adding nine to the previous term.
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This lets us know that whatever 64 plus nine is, that’s our next term.
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The next term in this sequence is 73.
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Here’s an example of a simple arithmetic sequence.
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We’re adding two each time.
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When we’re dealing with sequences, each of the terms also has a position.
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You can see the labels here.
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The two is in first position.
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The four is in second position.
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Six is in third position and then fourth and fifth, and so on.
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Positions are really important.
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Let’s take a closer look at what is happening here with these positions.
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Position one has a value of two.
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Position two has a value of four.
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To get from position one to position two, we need to add two.
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And to go from position two to position three, we need to add two.
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There is a relationship here between the position of a term and the term’s value.
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This column shows us that, to solve for position one, we multiply one times two.
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To solve for position two, the value, we take two.
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And we multiply it by two.
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To solve for position three, you multiply three by two to get six.
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In the operation, the first number we’re looking at is the position number.
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The two is the number that we’re adding to each term.
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Some of you might be wondering, Well why would I take the time to figure out the multiplication when I could just add two to six?
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Six plus two is eight.
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That’s pretty simple.
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And that’s true.
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If I wanted to find the next number in this sequence, I would probably just say 10 plus two is twelve.
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The sixth position is twelve.
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But if the question said something like what is the 80th position in this sequence, you don’t wanna add two 80 times.
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Now we’re going to add the number 80, the position 80, to our table.
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We would follow the same operation.
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In the 80th position of this sequence is the number 160.
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Finding that by multiplying 80 times two is significantly faster than trying to add two each time 80 times.
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You might also come across the question: what is an algebraic expression for finding terms in this sequence?
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In that case, we don’t know what position we’re looking for.
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We’re looking to write an expression that can be used for all positions.
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When dealing with positions and sequences, we usually use the letter 𝑛 to represent an unknown position.
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We could find the value of a term in position 𝑛 by multiplying the position by two, 𝑛 times two.
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We could say 𝑛 times two for our expression, or simply two 𝑛.
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We can plug in any position number here and find the term that would be in that position.
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We could plug in position 100, position seven, position 15.
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It doesn’t matter.
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This expression works for solving this sequence.
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Here’s another example.
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Find the next term in this sequence.
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51, 102, 153.
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What’s next?
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The first question we should ask is what is being added to each term.
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Last time, that was really easy because it was two.
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If you don’t immediately recognise what’s being added, here’s what you can do.
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You can take the term in the second position and subtract the term in the first position.
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102 minus 51 is 51.
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You could also subtract 102 from 53 or the position two number from the position three number.
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Both of these equal 51.
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51 is what is being added each time.
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To find the next term, we need to take the third term, 153, and add 51 to that.
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The next term in this sequence is 204.
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Here’s what a table for this sequence would look like.
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The operation here would be to take the position and multiply it by 51 because 51 is what we’re adding here.
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51𝑛 would be the expression that we could use to take any position and find its value.
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Here is our last example.
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Find the 70th term in the following sequence.
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Six, 12, 18, 24, ….
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We’re trying to answer the question what is being added to each term.
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And we know that here each term is six more than the previous term.
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But I’m not just trying to find out what are we adding to each term.
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I need to find out what expression can I use to find the 70th term.
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I notice the pattern is that you take the position.
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And you multiply it by six.
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So in order for me to find the 70th term, I need to multiply that by six.
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When I do that, I have a solution of 420.
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I also know that I can find any term in this sequence by taking the position 𝑛 and multiplying it by six.
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That expression is important.
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That’s all for this video.
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Now you can go try some sequences on your own.