Question 144811: Its about general terms. I dont get them at ALL!! like one practice problem says find the general term for 8,1,-6,-13,-20. Please help!!
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website! The general term of a series is a formula that you can use to calculate ANY term of the series without having to start at the beginning and calculate all of the intermediate terms first. In the problem you presented you are given the first five terms of a series that presumably goes on forever, namely 8, 1, -6, -13, -20... (note the elipsis at the end indicating that it goes on). Clearly, to get from one element in your series to the next, you must subtract 7 from the previous term. But how would you find the value of the 100th term without having to write out the 6th through the 99th term first? The trick is to find a relationship between the value of the term and the ordinal number of the term (ordinal numbers are 1st, 2nd, 3rd... etc.).
Your first term is 8, and we need to subtract zero 7s from 8 to get 8. So we can say that  is a way to represent the 1st term. The red colored 1 in the expression is because this is the first term.
The second term is 1, so we need to subtract one 7 from 8 to get 1. So we can say that  is a way to represent the 2nd term, and the red colored 2 is because this is the 2nd term.
The third term is -6. 8 minus 2 times 7 gives us -6, so 
And so on. Notice that the only thing that changes in each of these expressions is the ordinal number of the term we are calculating. Let  be the ordinal number of the general term, so the general term becomes  , which can be simplified:
So now if you wanted the 100th term, you could calculate it directly:
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