Find the common difference or ratio in each sequence. Write an algebraic expression for the sequence and find the next three terms.

6. 160, 40, 10, 2.5...

8. -1, 2, -4, 8....

Can you help me find the answers?

6: r = 1/4

8: r = -2

That should do it.

Yes, I can help you find the answers. To find the common difference or ratio in each sequence, we need to look at the pattern between the terms.

For the first sequence, we can divide each term by the previous term to see if there is a consistent ratio. Let's do that:

40/160 = 0.25
10/40 = 0.25
2.5/10 = 0.25

We can see that there is a consistent ratio of 0.25 between each term. Therefore, the common ratio in this sequence is 0.25.

To write an algebraic expression for this sequence, we can start with the first term (160) and then multiply it by the common ratio (0.25) repeatedly. Let's write the expression:

An = 160 * (0.25)^(n-1)

Now, to find the next three terms, we can substitute the values of n into the expression. Let's do that:

A2 = 160 * (0.25)^(2-1) = 160 * (0.25)^1 = 160 * 0.25 = 40

A3 = 160 * (0.25)^(3-1) = 160 * (0.25)^2 = 160 * 0.0625 = 10

A4 = 160 * (0.25)^(4-1) = 160 * (0.25)^3 = 160 * 0.015625 = 2.5

Therefore, the next three terms in the sequence are 40, 10, and 2.5.

Now, let's move on to the second sequence.

For the second sequence, we can subtract each term by the previous term to see if there is a consistent difference. Let's do that:

2 - (-1) = 3
-4 - 2 = -6
8 - (-4) = 12

We can see that there is a consistent difference of 3 between each term. Therefore, the common difference in this sequence is 3.

To write an algebraic expression for this sequence, we can start with the first term (-1) and then add the common difference (3) repeatedly. Let's write the expression:

An = -1 + 3(n-1)

Now, to find the next three terms, we can substitute the values of n into the expression. Let's do that:

A2 = -1 + 3(2-1) = -1 + 3(1) = -1 + 3 = 2

A3 = -1 + 3(3-1) = -1 + 3(2) = -1 + 6 = 5

A4 = -1 + 3(4-1) = -1 + 3(3) = -1 + 9 = 8

Therefore, the next three terms in the sequence are 2, 5, and 8.