32,48,72,108...Arithmetic or geometric I think it is arithmetic

Then Find common difference or ratio
multiple of 12?
What are the next 3 terms
120,132,144
What is the "zero" term
12 not sure what this means though
What is the equation?
12n * n

To determine whether the sequence 32, 48, 72, 108... is arithmetic or geometric, we can observe the differences between consecutive terms:

48 - 32 = 16
72 - 48 = 24
108 - 72 = 36

Since the differences are not the same, the sequence is not arithmetic, but it is geometric.

To find the common ratio, we can divide each term by the previous term:
48 / 32 = 1.5
72 / 48 = 1.5
108 / 72 = 1.5

The common ratio is 1.5.

To check if the terms are multiples of 12, we can divide each term by 12:
32 / 12 = 2.67 (not a multiple of 12)
48 / 12 = 4 (multiple of 12)
72 / 12 = 6 (multiple of 12)
108 / 12 = 9 (multiple of 12)

Only the third term is a multiple of 12.

To find the next three terms, we can multiply the previous term by the common ratio:
108 * 1.5 = 162
162 * 1.5 = 243
243 * 1.5 = 364.5

The next three terms are 162, 243, and 364.5.

It is not clear what is meant by the "zero" term. If you are referring to the first term in the sequence, it is 32.

The equation for this geometric sequence can be written as: an = 32 * (1.5)^(n-1), where n is the term number.

To determine whether the given sequence 32, 48, 72, 108... is arithmetic or geometric, we need to check if there is a constant difference or ratio between consecutive terms.

For an arithmetic sequence, the difference between consecutive terms is constant. Let's check the differences:
48 - 32 = 16
72 - 48 = 24
108 - 72 = 36

As none of the differences are equal, we can conclude that the sequence is not arithmetic.

To determine if it is geometric, we need to check if there is a constant ratio between consecutive terms.
Let's check the ratios:
48 / 32 = 1.5
72 / 48 = 1.5
108 / 72 = 1.5

Since all the ratios are equal (1.5), we can conclude that the sequence is indeed geometric.

Now, to find the common ratio in the sequence, we take any term and divide it by its previous term.
For example:
48 / 32 = 1.5

So, the common ratio in this geometric sequence is 1.5.

To find the next three terms in the sequence, we multiply each term by the common ratio:
Last term = 108
Next term = 108 * 1.5 = 162
Next term = 162 * 1.5 = 243
Next term = 243 * 1.5 = 364.5

Therefore, the next three terms in the sequence are: 162, 243, 364.5.

Regarding the term "zero" term, it seems to refer to the first term in the sequence. In an arithmetic sequence, this would be called the "initial term," while in a geometric sequence, it would be called the "first term." However, since the given sequence is neither arithmetic nor geometric, there is no well-defined "zero" term in this context.

Lastly, the equation that represents the sequence can be found by identifying the pattern in the terms. In this case, we can see that each term (T) can be obtained by multiplying 12 by the position of the term (n). Therefore, the equation that represents the sequence is:
T = 12n.

48 - 32 = 16 = d

108 - 72 = 36 = d

d is NOT CONSTANT SO NOT arithmetic !!!!!!!
but
48/32 = 1.5 = r
108/72 = 1.5 = r
SUCCESS ! r, the ratio, is 1.5, GEOMETRIC

term n = 32 (1.5)^(n-1)

when n = 1
term 1 = 32 (1.5)^(1-1) = 32(1.5)^0
but anything^0 = 1
so
first term is 32 when n-1 = 0

32 48 72 108 162 243 364.5 546.75 etc