For each of the following explicit descriptions of a sequence, find a recursive description; for the recursive ones, find an explicit rule. Assume sequences start from n = 1.

1. an = 2n2 − 3n
2. a1 = 4 and an+1 = an + 2
3. a1 = −8 and an+1 = an 2
4. an =3n

#1. quadratic sequences have a constant second difference. In this case, 4. The sequence and differences are

-1, 2, 9, 20, 35, ...
...3, 7, 11, 15, ...
.....4, 4, 4, ...

So, the 1st differences form an arithmetic progression: 3 + 4k

So,
a1 = -1
an+1 = an + 3+4(n-2)
or, an+1 = an + 4n-5

#2 Since each term is just 2 more than the last, it's just an AP. List the first few terms, and it's easy to see.

4,6,8,10,...
an = 2n+2

#3 has a typo. It's either
an+1 = an * 2 or n+1 = n^2

The first case is awkward due to the negative first term.

The second case is easy. Just keep multiplying by 2: an = -4*2^n

#4 is easy, since it just means that you keep adding 3. I'm sure you can handle that.

Looks like I had mismatched subscript tags. I meant

#1. quadratic sequences have a constant second difference. In this case, 4. The sequence and differences are

-1, 2, 9, 20, 35, ...
...3, 7, 11, 15, ...
.....4, 4, 4, ...

So, the 1st differences form an arithmetic progression: 3 + 4k

So,
a1 = -1
an+1 = an + 3+4(n-2)
or, an+1 = an + 4n-5

#2 Since each term is just 2 more than the last, it's just an AP. List the first few terms, and it's easy to see.

4,6,8,10,...
an = 2n+2

#3 has a typo. It's either
an+1 = an * 2 or n+1 = n^2

The first case is awkward due to the negative first term.

The second case is easy. Just keep multiplying by 2: an = -4*2^n

#4 is easy, since it just means that you keep adding 3. I'm sure you can handle that.

for #3 it should be an/2

To find the recursive description for each explicit description of a sequence, we need to find a rule that defines how to calculate each term in the sequence based on previous terms. Let's go through each sequence one by one:

1. Explicit description: an = 2n^2 - 3n
Recursive description: For n = 1, a1 = 2(1)^2 - 3(1) = -1
For n > 1, an = an-1 + (2n - 3)

2. Explicit description: a1 = 4 and an+1 = an + 2
Recursive description: For n = 1, a1 = 4
For n > 1, an = an-1 + 2

3. Explicit description: a1 = -8 and an+1 = an^2
Recursive description: For n = 1, a1 = -8
For n > 1, an = (an-1)^2

4. Explicit description: an = 3n
Recursive description: For n = 1, a1 = 3(1) = 3
For n > 1, an = an-1 + 3

Now, let's find the explicit rule for the recursive descriptions:

1. Recursive description: an = an-1 + (2n - 3)
Explicit rule: an = n^2 + n - 1

2. Recursive description: an = an-1 + 2
Explicit rule: an = 2n + 2

3. Recursive description: an = (an-1)^2
Explicit rule: an = (-8)^(2^(n-1))

4. Recursive description: an = an-1 + 3
Explicit rule: an = 3n

These are the recursive descriptions and explicit rules for each given sequence.