Tell whether each sequence is arithmetic. Justify your answer. If the sequence is arithmetic, write a function rule to represent it.

15. 128, 64, 32, 16...
16. 3, 3.25, 3.5, 3.75..

15. The given sequence is 128, 64, 32, 16...

To determine if this sequence is arithmetic, we can check if the difference between consecutive terms is constant.

Let's find the differences between consecutive terms:
64 - 128 = -64
32 - 64 = -32
16 - 32 = -16

The differences are not constant; they are decreasing by a factor of 2 each time. Therefore, the sequence is not arithmetic.

16. The given sequence is 3, 3.25, 3.5, 3.75...

To determine if this sequence is arithmetic, we can check if the difference between consecutive terms is constant.

Let's find the differences between consecutive terms:
3.25 - 3 = 0.25
3.5 - 3.25 = 0.25
3.75 - 3.5 = 0.25

The differences are constant; each difference is 0.25. Therefore, the sequence is arithmetic.

To write a function rule to represent the arithmetic sequence, we can use the formula: a_n = a_1 + (n-1)d, where a_n represents the nth term, a_1 represents the first term, n represents the position of the term, and d represents the common difference.

In this case, the first term (a_1) is 3, and the common difference (d) is 0.25.

So, the function rule to represent the given arithmetic sequence is:
a_n = 3 + (n-1)0.25

To determine whether a sequence is arithmetic, we need to examine the differences between consecutive terms. If the differences are constant, then the sequence is arithmetic.

Let's analyze each sequence:

15. 128, 64, 32, 16...

To find the differences between consecutive terms, we subtract each term from its preceding term:

64 - 128 = -64
32 - 64 = -32
16 - 32 = -16

The differences are not constant. In this case, the sequence is not arithmetic.

16. 3, 3.25, 3.5, 3.75...

Again, let's find the differences between consecutive terms:

3.25 - 3 = 0.25
3.5 - 3.25 = 0.25
3.75 - 3.5 = 0.25

The differences between consecutive terms are constant (0.25). Therefore, this sequence is arithmetic.

To write a function rule representing a arithmetic sequence, we can use the general form of an arithmetic sequence, which is:

an = a1 + (n-1)d

where:
an represents the nth term,
a1 represents the first term,
n represents the position of the term (starting from 1),
d represents the common difference.

For the given sequence (3, 3.25, 3.5, 3.75...), we can see that the first term (a1) is 3, and the common difference (d) is 0.25. Therefore, the function rule representing this sequence is:

an = 3 + (n-1)*0.25

check the differences:

#15: -64,-32,-16,...
#16: 0.25,0.25,0.25,...

Looks like #16 is

Tn = 2.75 + 0.25n