I am so lost on these problems.

Write a geometric sequence that starts with 3 and has a common ratio of 5. What is the 23rd term in the sequence.

Write an arithmetic sequence that has a common difference of 4 and the eighth term is 13. What is the first term? What is the 23rd term in the sequence.

Write an arithmetic sequence where 12 is the first term of 12, and the tenth term is 39. What is the first term? What is the 23rd term in the sequence.

Two consecutive terms of a geometric sequence, in order are 10 and 12. What is the ratio. What is the term before 10? What is the term after 12?

well, they have said that

a=3
r=5
The 23rd term is thus 3*5^22
That is, you start with 3 and multiply by 5 22 more times to get to the 23rd term.

Next, we have
a + 7*4 = 13
so, a = -15
T23 = -15 + 22*4

Any two consecutive terms of a geometric sequence have a common ratio. In this case, we have

r = 12/10 = 6/5
So, to get the term before 10, divide by 6/5
To get the term after 12, multiply by 6/5

To solve these problems, we need to understand the concepts of arithmetic sequences and geometric sequences.

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In other words, each term is obtained by adding a fixed value (called the common difference) to the previous term.

A geometric sequence is a sequence of numbers in which the ratio of any two consecutive terms is constant. In other words, each term is obtained by multiplying the previous term by a fixed value (called the common ratio).

Let's solve these problems step by step:

1. Geometric sequence with a common ratio of 5, starting with 3:
To find the 23rd term in the sequence, we can use the formula for the nth term of a geometric sequence, which is given by:
An = A * r^(n-1)
where An is the nth term, A is the first term, r is the common ratio, and n is the term number.

In this case, A = 3, r = 5, and n = 23. Plugging these values into the formula, we get:
A23 = 3 * 5^(23-1)
A23 = 3 * 5^22 (since 23-1 = 22)

Computing this expression will give us the 23rd term of the sequence.

2. Arithmetic sequence with a common difference of 4, and the eighth term is 13:
To find the first term, we can use the formula for the nth term of an arithmetic sequence, which is given by:
An = A + (n-1)d
where An is the nth term, A is the first term, d is the common difference, and n is the term number.

In this case, d = 4, n = 8, and An = 13. Plugging these values into the formula, we get:
13 = A + (8-1) * 4
13 = A + 7 * 4

Solving this equation will give us the first term of the arithmetic sequence. To find the 23rd term, we can again use the formula mentioned above, substituting n = 23.

3. Arithmetic sequence with a first term of 12, and the tenth term is 39:
Similar to the previous problem, we can use the formula for the nth term of an arithmetic sequence to find the first term and the 23rd term.

In this case, A = 12, n = 10, and An = 39. Plugging these values into the formula, we get:
39 = 12 + (10-1) * d
39 = 12 + 9 * d

Solving this equation will give us the common difference (d) and the first term of the arithmetic sequence. We can then find the 23rd term by substituting n = 23.

4. Two consecutive terms of a geometric sequence are 10 and 12:
To find the common ratio, we divide the second term by the first term. In this case, the common ratio is 12/10 = 1.2.

To find the term before 10, we divide 10 by the common ratio. In this case, the term before 10 is 10/1.2 (or 8.3333...).

To find the term after 12, we multiply 12 by the common ratio. In this case, the term after 12 is 12 * 1.2 (or 14.4).

By understanding and using these formulas and methods, you should be able to solve these problems and find the requested values in each sequence.