SOLUTION: The third term of an arithmetic sequence is -1 and the seventh term is -13; how would I find the explicit formula for this sequence, and what would a22 be?

In the arithmetic sequence (or arithmetic progression) , ,,... 



each  is related to the previous  by a common difference, d.



This means we can write the progression as:



, , , ..., , ...



You are given  = -1  and  









Subtracting the  expression from the  expression:

   --> d = -3



With the common difference, you can easily find  and then write



  for n = 1,2,3,...



If you plug in n=22 you will find 





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EDITed to fix the display of the arithmetic progression (commas inside curly braces seem to break things).

 

Between the third term and the seventh term of an arithmetic progression, there are exactly 4 gaps in the number line 

of equal sizes.


Hence, the common difference of the arithmetic progression (which is exactly the size of a single gap) is

     =  = -3.


Thus we just found that the common difference of the AP is -3.


Now it is easy to find its first term   =  - 2*(-3) = -1 + 6 = 5.


Finally,   =  + d*(22-1) = 5 + (-3)*21 = -58.    ANSWER