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Where Does the Formula for a Term in an Arithmetic Sequence Come From?

What is the formula for the nth term in an arithmetic sequence.

Summary

  1. The subscript 1 in a1 stands for the 1st term
  2. To get to the next term, just add 3, the common difference!
  3. an is the nth term
  4. Each term added to a1 is a multiple of the common difference, 3.
  5. Notice how the 3rd term is equal to 1 plus 2 times the common difference. 3-1=2, and that is the key to the pattern!
  6. (n-1)•d results in multiples of 3
  7. It's good to verify the formula we derived with a term later in the sequence.

Notes

    1. Making tables is a really good technique for finding patterns!
    2. a1 is the 1st term
    1. Difference is from one term to the next
    2. a1 is the 1st term
    3. a2 is the next term
    4. 'd' is the common difference
    1. Each term an = a1+something=1+something
    1. The common difference, d=3
    2. Each term an = a1+something •3
    1. a1 is the 1st term
    2. a2 is the 2nd term
    3. 3=d, the common difference
    4. To get the 2nd term we multiply the common difference by 1, to get the 3rd term we multiply the common difference by 2. Do you see the pattern?
    1. an = a1+(n-1)•d