2.11 C) Arithmetic Sequences: Position to Term Rule – Part 2
We are now going to have a look at a few sequences and place them into the nth term rule. A sequence in the nth term rule is something like:
The coefficient of n is the difference between consecutive terms in the sequence. For the above example, the coefficient of n is 7, which means that the difference between terms is 7. Another way of viewing this is that we find the next term by adding 7 to the previous term. The constant (the number by itself), which in the above example is 4, allows us to get to the starting point of the sequence. We obtain the value for the constant (number) by finding what the 0th term (we take the difference off the first term).
When we are placing a sequence in the nth term rule, we should find the difference first (which will give you the coefficient of n) and then find what the constant (number) will be. Let’s now have a few examples.
Write the sequence below in the nth term rule.
The first step in placing this sequence into the nth term rule is to find out what the difference is between each term and the term that precedes it.
The difference between terms is always 2, which means that the coefficient of n is 2.
We now need to find what the constant is. We do this by finding what the 0th term is, which we find by taking the difference from the first term.
The 0th term would be 1. Therefore, this sequence can be written as 2n + 1.
Whenever we have found the sequence in the nth term rule, it is a good idea to check whether we are correct, and we can do this by checking a certain term in the sequence. Let’s check that our nth term rule gives us the correct value for the fourth term (which is 9). The value of n for the fourth term is 4, so we can check that we have the correct nth term rule by subbing in n as 4.
The nth term rule does give us 9, which means that we have the correct nth term formula; the nth term rule is 2n + 1.
Work out the nth term rule for the sequence below and find the 20th and 35th term.
The first step in placing this sequence into the nth term rule is to find what the difference is between a term and the term before it. The common difference between the terms is -6. This means that the coefficient of n is -6.
We now need to find what the constant is and we do this by finding what the 0th term is. We find the 0th term by taking the difference away from the first term; 35 – (-6) = 35 + 6 = 41. Therefore, the 0th term is 41 and this is the constant (number) in the nth term rule for this sequence.
The nth term rule for this sequence is -6n + 41.
It is a good idea to check that we have found the nth term rule correctly by checking that a term in the sequence is the same number that we would get if we subbed the value for n for that term into the nth term rule. I am going to check the fourth term, which is 17. The fourth term has a value for n of 4, so we sub n as 4 into the nth term rule.
The nth term rules states that the fourth term is 17, which is the same as the fourth term in the sequence, thus implying that we have found the correct nth term rule. We now need to find the 20th and 35th term, and we do this by subbing n as 20 for the 20th term and n as 35 for the 35th into the nth term rule.
Therefore, the 20th term in this sequence is -79 and the 35th term is -169.