Solve the following equation and show all of your steps: x^2+x−1=0
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An arithmetic sequence
is one in which consecutive terms of the sequence differ by a fixed number, call it <em>d</em>. This means that, given the first term , we can build the sequence by simply adding <em>d</em> :
and so on, the general pattern governed by the recursive rule,
We can exploit this rule in order to write any term of the sequence in terms of the first one. For example,
and so on up to
In this case, we're not given the first term right away, but the 17th. But this isn't a problem; we can use the same exploit to get
and so on, up to the next term we know,
(Notice how the subscript of <em>a</em> on the right and the coefficient of <em>d</em> add up to the subscript of <em>a</em> on the left.)
The 28th term is -73, so we can solve for <em>d</em> :
To get the first term of the sequence, we use the rule found above and either of the known values of the sequence. For instance,
Then the recursive rule for this particular sequence is