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2.11 A) Arithmetic Sequences – Introduction
2.11 A) Arithmetic Sequences – Introduction
Sequences are sets of numbers that follow a rule or pattern. We are going to be looking at lots of different sequences in this section. Two of the sequences that we are going to look at are arithmetic sequences and geometric sequences.
Arithmetic sequences are when the same number is added or subtracted to the previous term each time. Each term in an arithmetic series differs from the preceding one by a constant value.
Geometric sequence are where we multiply or divide the previous term by the same number each time to obtain the next term in the series.
Arithmetic sequences are when the same number is added or subtracted to the previous term each time. Each term in an arithmetic series differs from the preceding one by a constant value.
Geometric sequence are where we multiply or divide the previous term by the same number each time to obtain the next term in the series.
Arithmetic Sequences
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. An example of an arithmetic sequence is:
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. An example of an arithmetic sequence is:
We can see that the difference between each of the terms and their preceding term is 4. Therefore, in this sequence, we are adding 4 to each term to obtain the next term in the sequence.
The next 3 terms in this sequence would be 22 (18 + 4), 26 (22 + 4) and 30 (26 + 4).
We can find out what the pattern in a sequence is by using the term to term rule or position to term rule.
Term to Term Rule
The term to term rule tells us what we have to do to get from one term to the next term. We are going to describe the sequences in the next two examples by using the term to term rule.
The term to term rule tells us what we have to do to get from one term to the next term. We are going to describe the sequences in the next two examples by using the term to term rule.
Example 1
Work out the next 3 terms in the following sequence:
Work out the next 3 terms in the following sequence:
We work out the term to term rule by finding out what happens between each of the terms. We do this by subtracting the terms from the term that precedes it (for example, we would find the difference between 5 and 8 by doing 8 – 5, which is 3). When we do this for the rest of the terms, we see that the difference between the terms is 3.
We find the next 3 terms in the sequences by adding 3 onto the previous term. The next three terms in the sequence is 20 (17 + 3), 23 (20 + 3) and 26 (23 + 3). The original sequence with the next 3 terms in is:
We can describe the sequence in a term to term rule by giving the starting number and what the rule is (which is what the common difference is between each term). For our sequence, the starting number is 5 and rule is to add 3.
Example 2
Work out the next 3 terms in the following sequence:
Work out the next 3 terms in the following sequence:
Like before, we work out what the difference is between the terms and we do this by subtracting a term from the term that precedes it. It is a good idea to check a few of the differences to see that the difference is constant between the terms. The difference between the second and first term is -1.5 (15 – 13.5), the difference between the third and second term is -1.5 ( 12 – 13.5) and the difference between the fourth and third term is also -1.5 (10.5 – 12). Therefore, the difference is the same between all of the terms and it is -1.5. This means that we can describe the sequence as starting at 15 and the rule is that we subtract 1.5.
We now need to obtain the next three terms. We obtain the fifth term by taking 1.5 from the fourth term, which results in the fifth term being 9 (10.5 – 1.5). The sixth term is obtained by taking 1.5 from the fifth term, which means that the sixth term is 7.5 (9 – 1.5). And finally, the seventh term is obtained by taking 1.5 from the sixth term, which results in the seventh term being 6 (7.5 – 1.5). The original sequence with the next 3 terms is given below: