1. Give a recursive definition

Hello! i am stuck on this recursive definition. I don't know what to do next.

Q: Give a recursive definition of the sequence {an}, n = 1, 2, 3, … if
a.) an = 4n – 2
b.) an = 1 + (-1)^(n)
c.) an = n(n + 1)
d.) an = n²
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I have this:

a.)

an = 4n – 2
a1 = 4(1) – 2
a1 = 4 – 2
a1 = 2

an = 4n – 2
a2 = 4(2) – 2
a2 = 8 - 2
a2 = 6

an = 4n – 2
a3 = 4(3) – 2
a3 = 12 - 2
a3 = 10

an = 4n – 2
a4 = 4(4) – 2
a4 = 16 - 2
a4 = 14

I can see they go up by 4 for part a, but do i leave it like this?

b.)

an = 1 + (-1)^(n)
a1 = 1 + (-1)^(1)
a1 = 1 – 1
a1 = 0

an = 1 + (-1)^(n)
a2 = 1 + (-1)^(2)
a2 = 1 + 1
a2 = 2

an = 1 + (-1)^(n)
a3 = 1 + (-1)^(3)
a3 = 1 – 1
a3 = 0

an = 1 + (-1)^(n)
a4 = 1 + (-1)^(4)
a4 = 1 + 1
a4 = 2

I can see the sequence going 0, 2, 0, 2, ... etc.

c.)

an = n(n + 1)
a1 = 1(1 + 1)
a1 = 1(2)
a1 = 2

an = n(n + 1)
a2 = 2(2 + 1)
a2 = 2(3)
a2 = 6

an = n(n + 1)
a3 = 3(3 + 1)
a3 = 3(4)
a3 = 12

an = n(n + 1)
a4 = 4(4 + 1)
a4 = 4(5)
a4 = 20

I am not sure on this one?

2. For part a, the sequence is $2,\;6,\;10,\;...$.
$a_1 =2$ and $n \geqslant 2\;,\,a_n = a_{n - 1} + 4$.

Now you show us part b.

3. Plato, I don't understand this:

and .

I got the sequence of 2, 6, 10, etc. But i don't understand the above part of what you wrote?

4. Originally Posted by Grillakis
Plato, I don't understand this:

and .

I got the sequence of 2, 6, 10, etc. But i don't understand the above part of what you wrote?
Are you saying that you do not understand what a recursively defined sequence is?
If so, why are you asked to do this problem?

5. Well reason why im doing this is because its part of my HW and I missed his class when he supposedly covered this. I was sick so i am left to understand this on my own. He posts his lecture notes online and that is what I am going over. When I am looking through the textbook I am trying to understand it and when i can't I usually search online and I try to understand that this way or through forums, hopefully trying to get some help on how to do this. And if this isn't bad enough, I have been trying to learn the material on my own b/c my professor has a thick accent and its hard to understand him.

6. Look at this sequence: $a_1 =0$ then if $n\ge2$ define $a_n = 2 - a_{n-1}$.
Now you write out the next six terms. What do you see?

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give me a definition of sequence

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