Final answer:
The nth term of the sequence 192, -144, 108, ... is found using the formula of a geometric sequence. The first term is 192, and the common ratio is -0.75. Thus, the nth term can be calculated as 192 x (-0.75)^(n-1).
Step-by-step explanation:
To find the formula for the nth term of the given sequence 192, -144, 108, ... we notice a pattern. We can see that each term is being multiplied by a constant ratio to get to the next term. This is characteristic of a geometric sequence, where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
The first term of the sequence (a1) is 192, and to find the common ratio (r), we can divide the second term by the first term:
-144 ÷ 192 = -¾ or -0.75
Now, the nth term (an) of a geometric sequence can be found using the formula:
an = a1 × rn-1
Applying this formula, the nth term of the given sequence is:
an = 192 × (-¾)n-1
Or expressed as a decimal:
an = 192 × (-0.75)n-1