Write a rule for the sequence: 3, 9, 27, 81, ...

Find the next three terms of the sequence: 2, 5, 8, 11, ...
What is the 10th term in the following sequence? 2, 5, 8, 11, ...
Which explains why the sequence 1, 2, 4, 8, ... is arithmetic or geometric?

To find the rule for the sequence 3, 9, 27, 81, ..., it is helpful to observe the pattern. Each term is obtained by multiplying the previous term by 3:

3 x 3 = 9
9 x 3 = 27
27 x 3 = 81

So, the rule for this sequence is to multiply each term by 3. Therefore, the next term in the sequence would be 81 x 3 = 243. The following two terms would be 243 x 3 = 729, and 729 x 3 = 2187.

For the sequence 2, 5, 8, 11, ..., it is evident that each term increases by 3. This means that the rule for this sequence is to add 3 to each term. So, the next three terms would be 11 + 3 = 14, 14 + 3 = 17, and 17 + 3 = 20.

To find the 10th term in the sequence 2, 5, 8, 11, ..., we can identify the rule as adding 3 to each term. Starting from the first term, we can follow this rule and add 3 to each subsequent term until we reach the 10th term:

2, 5, 8, 11, 14, 17, 20, 23, 26, 29

Therefore, the 10th term in the sequence is 29.

The sequence 1, 2, 4, 8, ... is a geometric sequence. This is because each term is obtained by multiplying the previous term by 2:

1 x 2 = 2
2 x 2 = 4
4 x 2 = 8

Since each term is obtained by multiplying the previous term by the same value (in this case, 2), the sequence is geometric.

For the sequence 3, 9, 27, 81, ..., the rule can be described as multiplying each term by 3 to obtain the next term. So, the next three terms would be 243, 729, 2187.

For the sequence 2, 5, 8, 11, ..., the rule can be described as adding 3 to each term to obtain the next term. So, the next three terms would be 14, 17, 20.

To find the 10th term in the sequence 2, 5, 8, 11, ..., we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the position of the term in the sequence, and d is the common difference. In this sequence, the first term a1 is 2, the common difference d is 3, and n is 10. Substituting these values into the formula, we have a10 = 2 + (10 - 1)3 = 2 + 9(3) = 2 + 27 = 29. Therefore, the 10th term is 29.

The sequence 1, 2, 4, 8, ... is geometric because each term is obtained by multiplying the previous term by a constant value. In this case, multiplying each term by 2 gives the next term.