SOLUTION: What is the 20th term of the sequence that begins −3, 6, −12, 24…?

First we check to see whether it's an arithmetic 
sequence or a geometric sequence or neither:

Check to see if it's an arithmetic sequence:

2nd term - 1st term = 6 - (-3) = 6+3 = 9
3rd term - 2nd term = -12 - 6 = - 18
4th term - 3rd term = 24 - (-12) = 24 + 12 = 36

No, because if it were those would all be the same and 
that would be the common difference d.  So it's not an 
arithmetic sequence.

Check to see if it's a geometic sequence:

2nd term ÷ 1st term = 6 ÷ (-3) = -2
3rd term ÷ 2nd term = -12 ÷ 6 = -2
4th term ÷ 3rd term = 24 ÷ (-12) = -2

Yes, it's a geometric sequence because those are all the same,
and -2 is the common ratio r.

So we use the formula for the nth term of a geometric sequence,
which is:

an = a1r(n-1)

where an is the nth term, so n = 20,
a1 is the first term, which is -3, 
and r is the common ratio, which is -2. 

a20 = -3(-2)(20-1)

a20 = -3(-2)19

a20 = -3(-2)19

a20 = -3(-524288)

a20 = 1572864

Checking:

Term        
no.          term
1             -3
2             6
3             -12
4             24
5             -48
6             96
7             -192
8             384
9             -768
10            1536
11            -3072
12            6144
13            -12288
14            24576
15            -49152
16            98304
17            -196608
18            393216
19            -786432
20            1572864

Edwin