If the fifth and eighth terms of an arithmetic sequence are minus 9 and minus 21, respectively, what are the first four terms of the sequence?

One way to do this problem is to determine the common difference.  If the 5th and 8th terms are -9 and -21, we can do this by subtracting -9 from -21:

-21-(-9) = -12.  The 5th and 8th terms are not consecutive, so we have to think in terms of (8-5), or 3, times the common difference to get from -9 to -21.

Note that -12 divided by 3 is -4.  Thus, the common difference is -4.

Check:  -9 - 4 = -13; -13 - 4 = -17; -17 - 4 = -21 (which is correct).

We know that the 5th term is -9.  To find the 4th term, work backwards:  subtract (-4) from -9, which produces -9+4=-5.

The fourth term is -5.  Subtracting -4 from this (which is the same as adding 4 to this -5) produces the third term; it is -1.  Can you now find the 2nd and 1st terms using the same approach?

Don’t worry about it