SOLUTION: Write a possible explicit rule for the nth term of the sequence. Then find the 20th term. (5, -10, 20, -40, 80, -60) *I think the answer for the 20th term is -2,621,440 but i'm no

The odd numbered term go 5,20,80,320,1280,5120,20480. A geometric
sequence with first term 5 and common ratio 4.  That has 
explicit rule for the nth term:

 for n=1,3,5,7,9,...

Now we find the pattern for the even numbered terms:

Let's make a table of the even numbered terms.  We can divide all the 
even numbered terms by -10, so we'll make a column of the terms
divided by -10.  Then we'll make a table of the amount we added to
the preceding even term.

                         Amt. added
                            to
Term                     preceding
 no.  Term  Term/(-10)   Term/(-10)
 2.   -10     1             ---
 4.   -40     4              3
 6.   -60     6              2 
 8.
10.
12.
14.
16.
18.
20.

Aha!  That last column starts with 3,2, so that seems to go 

3,2,1,0,-1,-2,-3,-4,...

So we fill those numbers in the last column and work backward:

                         Amt added
                            to
term                     preceding
 no.  term  term/(-10)   term/(-10)
 2.   -10     1             ---
 4.   -40     4              3
 6.   -60     6              2 
 8.   -70     7              1
10.   -70     7              0
12.   -60     6             -1
14.   -40     4             -2
16.   -10     1             -3
18.   +30    -3             -4
20.   +80    -8             -5

So using the pattern of extending 3,2 in the last column
to 3,2,1,0,-1,-2,..., then working backward, we get that 
the 20th term is 80.

We can find a quadratic explicit rule for the even terms
by assuming the formula:



Substituting n=1,2,3 forming a system of 3 equations in
3 variables, solving getting , , and , for n = 2,4,6,8,...

Edwin