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2.11 B) Arithmetic Sequences: Position to Term Rule - Part 1
2.11 B) Arithmetic Sequences: Position to Term Rule - Part 1
Arithmetic sequences are where each term in the sequence differs from the preceding term by a constant value. This value can be positive or negative.
There are two different methods that can be used to understand what is happening with an arithmetic sequence. These two methods are the term to term rule and the position to term rule. The section before looked at how to use the term to term rule and in this section, we are going to be looking at the position to term rule.
There are two different methods that can be used to understand what is happening with an arithmetic sequence. These two methods are the term to term rule and the position to term rule. The section before looked at how to use the term to term rule and in this section, we are going to be looking at the position to term rule.
Position to Term Rule
The position to term rule is also known as the nth term. It uses algebra to quickly work out what a term is if the position of the term in the sequence is known. We denote the position of the term in the sequence as n. The value for n for the first term is 1, the value for n for second term is 2, third term is 3, fourth term is 4, … fiftieth term is 50 etc.
An example of an arithmetic sequence that is written in the position to term rule is 4n + 1. We are able to find specific terms in the sequence by subbing in various values for n.
Example 1 – Using the Rule
For the sequence 4n + 1, find the first, second, ninth and fiftieth term.
To find the values for these terms, we will be subbing in the various values for n.
First Term
The value for n for the first term is 1. Therefore, we find the first term by subbing in n as 1.
For the sequence 4n + 1, find the first, second, ninth and fiftieth term.
To find the values for these terms, we will be subbing in the various values for n.
First Term
The value for n for the first term is 1. Therefore, we find the first term by subbing in n as 1.
This means that the first term in the sequence is 5.
Second Term
We find the value for the second term by subbing in n as 2.
Second Term
We find the value for the second term by subbing in n as 2.
The second term is 9. When we are given a sequence in the term to term rule, the coefficient of n is the difference between consecutive terms. The sequence that we are working with is 4n + 1, and the coefficient of n is 4, which means that we obtain the next term by adding 4 to the preceding term. We can see that this is the case by looking at the first two terms that we have found. The first term is 5 and the second term is 9; the difference between these two terms is 4 (the coefficient of n).
Ninth Term
Now let’s find the ninth term. The value of n for the ninth term is 9, so we can find the value of the ninth term by subbing in n as 9.
Ninth Term
Now let’s find the ninth term. The value of n for the ninth term is 9, so we can find the value of the ninth term by subbing in n as 9.
The ninth term is 37.
Fiftieth Term
We now need to find the fiftieth term. The value of n for the fiftieth term is 50, so we obtain the fiftieth by subbing in n as 50.
Fiftieth Term
We now need to find the fiftieth term. The value of n for the fiftieth term is 50, so we obtain the fiftieth by subbing in n as 50.
The fiftieth term in the sequence is 201.
Example 2
A sequence has the following term to term rule:
A sequence has the following term to term rule:
Find the fourth and ninth term.
Fourth Term
We find the fourth term by subbing n in as 4 into the term to term rule.
Fourth Term
We find the fourth term by subbing n in as 4 into the term to term rule.
The fourth term is 5.
Ninth Term
We find the ninth term by subbing n in as 9 into the term to term rule.
Ninth Term
We find the ninth term by subbing n in as 9 into the term to term rule.
The ninth term is 20.
Example 3
A sequence has the following term to term rule:
A sequence has the following term to term rule:
Find the 6th and 49th term.
6th Term
We find the 6th term by subbing n in as 6 into the term to term rule.
6th Term
We find the 6th term by subbing n in as 6 into the term to term rule.
The 6th term is -8.
49th Term
We find the 49th term by subbing n in as 49 into the term to term rule.
49th Term
We find the 49th term by subbing n in as 49 into the term to term rule.
The 49th term is -223.