Question

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

Answer 1

#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,
a<sub>₁<sub> = -1
a<sub>n+1<sub> = a<sub>_n<sub> + 3+4(n-2)
or, a<sub>n+1<sub> = a<sub>_n<sub> + 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,...
a<sub>_n<sub> = 2n+2

#3 has a typo. It's either
a<sub>_n+1<sub> = a<sub>_n<sub> * 2 or <sub>_n+1<sub> = <sub>_n<sub>^2

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

The second case is easy. Just keep multiplying by 2: a<sub>_n<sub> = -4*2^n

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

Answer 2

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,
a_₁ = -1
a_{_n+1} = a_{_n} + 3+4(n-2)
or, a_{_n+1} = a_{_n} + 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,...
a_{_n} = 2n+2

#3 has a typo. It's either
a_{_n+1} = a_{_n} * 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: a_{_n} = -4*2^n

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

Answer 3

for #3 it should be an/2