For the sequence, find a formula for the general term and specify the appropriate starting value of n.
-1, 2, -4, 8, -16,...
If we ignore the signs we see the sequence
1, 2, 4, 8, 16,...
These are the powers of 2 starting at n = 0. To account for the signs we need to multiply by (-1)n + 1. Starting at n = 0, the general term is
an = (-1)n + 12n.
If we start instead at n = 1 the general term is
an = (-1)n2n – 1.
If we start at n = 1 the numerator is n and the denominator is 5. The general term is . There's no reason to start at n = 0 because that would only make things more complicated.
If we look at the numerator first we see the sequence n! starting at n = 1. Then the denominator is 2n – 1. Starting at n = 1, the general term is
If instead we look at the denominator first we might see the sequence (2n + 1) starting at n = 0. Then the numerator is (n + 1)!. Starting at n = 0, the general term is
0, 3, 6, 9, 12,...
This is the sequence an = 3n starting at n = 0.
If we look at the denominator first we see the sequence (2n)!, starting at n = 1. Taking signs into account, the general term is
If we look at the signs first we might see the sequence (-1)n starting at n = 0. Then the general term is
Be Careful: When dealing with factorials, be careful with your parentheses. In particular,
(2n)! ≠ 2n!
For example, if n = 4 then
(2n)! = 8! = 40320
2n! = 2(4!) = 48.
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