2.11 E) Quadratic Sequences: Using the Nth Term Rule
Quadratic sequences have terms in them that contain squares. An example of a quadratic sequence is:
In order to find a certain term in a sequence, such as the 5th or 39th term, we sub in the respective values for n in exactly the same way as we did with the sequences that we looked at in the previous sections; we just need to make sure that we square the n that we are subbing into the nth term rule.
Find the 3rd, 4th and 9th term in the sequence below:
We find the 3rd term by subbing in n as 3, the 4th term by subbing in n as 4 and finally the 9th term by subbing in n as 9. When we are working with the first part of the nth term rule (the 2n2), we need to be careful to follow the rules of BODMAS/ BIDMAS; we need to square n first (other in BODMAS or indices in BIDMAS) and then multiply it by 2.
To find the 3rd term, we sub in n as 3.
Therefore, the 3rd term in the sequence is 23.
To find the 4th term in the sequence, we sub in n as 4.
The 4th term in the sequence is 37.
To find the 9th term in the sequence, we sub in n as 9.
Therefore, the 9th term in the sequence is 167.
Find the 5th and 11th term in the sequence below.
The highest power of n in the nth term formula above is 2. This means that this is a quadratic sequence. The nth term rule for this sequence is slightly more complex than the nth term rule for the sequence in example 1 and this is because we have 3 components in the nth term rule. However, we find the 5th and 11th term in exactly the same way as earlier; we sub in the respective value for n into the nth term rule.
In order to find the 5th term, we sub in n as 5 into the nth term rule.
Therefore, the 5th term is 17.
In order to find the 11th term, we sub in n as 11 into the nth term rule.
Therefore, the 11th term in the sequence is 95.
Quadratic sequences have unequal first difference. What I mean by this is that the difference between the terms changes each time. The differences between the differences is the same. This will all make much more sense when we have a look at an example.
Example 3
Find the next 3 terms in the sequence below.
This means that the difference between the fourth and fifth term will be 9 because it will be the difference between the third and fourth term (7) plus 2; 7 + 2. Therefore, the fifth term is 26 (17 + 9).
Find the next 3 terms in the sequence below.