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First we define an arithmetic sequence as one where each successive term has a common difference and that difference is constant. An example might be 1, 4, 7, 10, 13, 16, ..where the difference is 3. 1+3=4, 4+3=7 etc.

Here is a common example that is given as a problem but shows a real life example of arithmetic sequences.

A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater?

The common difference is 8 and we want the the sum of the first 20 terms this gives us the sum of all the seats. We solve this by first finding the 20th term which is 212 and noting that the first term is 60. We add the first and the 20th terms in the sequence and multiply the sum by 20. Next we divide that product by 2.

The sum we are looking for is 20(60+212)/2=2720 so there are 2720 seats in the theater!

The general formula to find the sum of the first n terms in an arithmetic sequence is to multiply n by the sum of the first and nth terms in the sequence and divide that answer by 2. In symbols we write Sn=n(a1+ an)/2

Q: How do you use arithmetic sequences in real life?

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The answer depends on the sequence. Different sequences may be used in different circumstances.

Of course! People were doing real arithmetic long before the first computer!

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You can use a calculator to actually add them up.Or you can use the formula for the sum of an arithmetic sequence - this has the advantage that you can also use it for larger sequences (for example, all even numbers up to a thousand).

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