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2.11 D) Arithmetic Sequences: Checking Whether a Number is in a Sequence
2.11 D) Arithmetic Sequences: Checking Whether a Number is in a Sequence
Sometimes it be the case in the exam that you are given the nth term formula for a sequence and are then asked whether a number is or is not in the sequence. In order to check whether a term is or is not in the sequence, we set the nth term formula equal to the number that may or may not lie in the sequence. We then solve to find the value of n. If the value of n that we obtain is an integer, the term does lie in the sequence. If the value of n that we obtain is not an integer, the term does not lie in the sequence. Also, if the value of n is an integer, the value of the integer tells us what position the term has in the sequence. For example, if we found the value of n to be 6, the term that we were checking would be the sixth term in the sequence.
This will all make much more sense when we work through a few examples.
This will all make much more sense when we work through a few examples.
Example 1
We have the sequence:
We have the sequence:
a) Is 23 in the sequence? If it is in the sequence, what term is it?
b) Is 825 in the sequence? If it is in the sequence, what term is it?
Part a
In order to answer part a of this question, we set the nth term formula equal to 23. This gives us the equation.
b) Is 825 in the sequence? If it is in the sequence, what term is it?
Part a
In order to answer part a of this question, we set the nth term formula equal to 23. This gives us the equation.
The next step is to solve for n. We do this by getting all of the terms that contain n to one side of the equation and all of the numbers to the other side. There are more n’s on the left side of the equation. Therefore, I am going to get all of the n’s to the left side of the equation and all of the numbers to the right. This means that we need to move the 3 that is currently on the left to the right, and we do this by doing the opposite; we take 3 from both sides.
We now have all of the n’s on the left and the numbers on the right. We want to find the value of n and not 4n. Therefore, we need to divide both sides by the coefficient of n; we divide both sides by 4.
This tells us that n is 5. This is an integer, which means that 23 is in the sequence and is it the fifth term.
Part b
Part b of the question asks us whether 85 is in the sequence. Like before, the first step is to set the nth term formula equal to 82. This gives us the equation below.
Part b
Part b of the question asks us whether 85 is in the sequence. Like before, the first step is to set the nth term formula equal to 82. This gives us the equation below.
The next step is to find the value of n. Like the question before, we need to get all of the n’s to one side and all of the numbers to the other side. Therefore, we need to move the 3 from the left side of the equation to the right side of the equation, which we do by taking 3 from both sides.
We want to find the value for n and not 4n, so we divide both sides of the equation by 4.
20.5 is not an integer and this means that 85 is not in the sequence.
Example 2
We have the sequence 39 – 6n.
a) Is 12 in the sequence? If it is in the sequence, what term is it?
b) Is -93 in the sequence? If it is in the sequence, what term is it?
Part a
We answer these questions in the exactly the same way as before; we set the nth term formula equal to the number that may or may not be in the sequence. This means that for part a, we get the equation that is given below.
We have the sequence 39 – 6n.
a) Is 12 in the sequence? If it is in the sequence, what term is it?
b) Is -93 in the sequence? If it is in the sequence, what term is it?
Part a
We answer these questions in the exactly the same way as before; we set the nth term formula equal to the number that may or may not be in the sequence. This means that for part a, we get the equation that is given below.
We want to find the value of n. We do this by getting all of the terms that contain n to one side of the equation and all of the numbers to the other side. We get the n’s to the side that currently has more n’s. This is the right side because there are more n’s on the right (0n) than on the left (-6n). This means that we need to move the -6n from the left side of the equation to the right, and we do this by adding 6n to both sides.
The next step is to move the 12 from the right side of the equation to the left, which we can do by taking 12 from both sides of the equation.
We want to find the value of n and not 6n, which means that we divide both sides of the equation by 6.
This tells us that n is 4.5. This is not an integer, which means that 12 does not lie in the sequence.
Part b
Part b asks us to check whether -93 is in the sequence. We check whether it is in the sequence by setting the nth term formula equal to -93. This gives us the equation below.
Part b
Part b asks us to check whether -93 is in the sequence. We check whether it is in the sequence by setting the nth term formula equal to -93. This gives us the equation below.
In order to find the value of n, we need to get all of the n’s to one side of the equation and all of the numbers to the other side. There are more n’s on the right side of the equation, so we get all of the terms that contain n on the right and all of the numbers on the left. Let’s start by moving the -6n from the left side of the equation to the right. We are able to move -6n by doing the opposite; we add 6n to both sides.
The next step is to move the -93 from the right to the left, which we do by doing the opposite; we add 93 to both sides.
We want to find the value of n and not 6n. Therefore, we divide both sides of the equation by 6.
Therefore, n is 22. This is an integer, which means that -93 is in the sequence and it is the 22nd term.