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How Do You Determine if a Sequence is Arithmetic or Geometric?

Determine whether each sequence is arithmetic or geometric:
a) -6, 1, 8, 15, 22
b) 324, 108, 36, 12, 4

Summary

  1. A sequence is a set of numbers in a particular order
  2. The common ratio is the number that a term is multiplied by to get the next term in a geometric sequence
  3. 'Diff' stands for 'difference'
  4. The common difference is the number that is added to a term to get the next term in an arithmetic sequence
  5. In Example a), we add 7 each time to get the next term, so we have an arithmetic sequence
  6. In Example b), we multiply by 1/3 each time to get the next term, so we have an geometric sequence

Notes

    1. The common ratio is the number that a term is multiplied by to get the next term in a geometric sequence
    2. If that number is the same for the whole sequence, then we have a geometric sequence
    1. In order for it to be a geometric sequence, that number needs to be the same for the whole sequence
    1. Remember, a ratio is just a comparison of numbers by division
    1. Find the ratio of the first two terms by taking the second term, 1, divided by the term before it, -6
    2. So 1/-6
    3. 'r' stands for 'ratio'
    4. The ratio of the first two terms is -1/6
    5. The next two consecutive terms are 1 and 8
    6. The ratio of these terms is 8
    1. Since -1/6 and 8 are not the same number, this sequence does not have a common ratio
    2. That means that it is not a geometric sequence
    1. If a sequence has a common difference, then it is an arithmetic sequence
    1. The difference is the amount of change between numbers in a sequence
    1. To find the difference between consecutive terms, take a term and subtract the one that comes before it
    2. The first two terms are -6 and 1
    3. Find the difference by taking 1-(-6)
    4. Subtracting a negative is the same as adding a positive, so 1-(-6) is the same as 1+6, which is 7
    5. 'd' stands for 'difference'
    6. So the difference between the first two terms is 7
    7. The next two consecutive terms are 1 and 8
    8. Their difference is 7 as well!
    1. In order to have a common difference, each pair of consecutive terms in the sequence needs to have that same difference of 7
    1. In order to have a common difference, each pair of consecutive terms in the sequence needs to have that same difference of 7
    2. We can add 7 to each term and see if we get the next term in the sequence
    1. The difference between consecutive terms in the sequence is 7
    2. So 7 is the common difference
    3. Since this sequence has a common difference, it's an arithmetic sequence!
    1. The common ratio is the number that each term is multiplied by to get the next term in a geometric sequence
    2. If that number is the same for the whole sequence, then we have a geometric sequence
    1. The first two terms in the sequence are 324 and 108
    2. Find the ratio by taking the second term divided by the first term
    3. 'r' stands for 'ratio'
    4. Simplifying the ratio, we get 1/3
    5. The next two consecutive terms are 108 and 36
    6. Their ratio also simplifies to 1/3
    1. To be sure this is a geometric sequence, we need to check the entire sequence
    2. If each set of consecutive terms has the same ratio, then this is a geometric sequence
    1. We can multiply each term by 1/3 and see if we get the next term in the sequence
    1. So 1/3 is the common ratio
    2. Since this sequence has a common ratio, it's a geometric sequence