# Give a recursive definition

• Mar 23rd 2009, 01:17 PM
Grillakis
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 = nē
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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?
• Mar 23rd 2009, 01:34 PM
Plato
For part a, the sequence is \$\displaystyle 2,\;6,\;10,\;...\$.
\$\displaystyle a_1 =2\$ and \$\displaystyle n \geqslant 2\;,\,a_n = a_{n - 1} + 4\$.

Now you show us part b.
• Mar 23rd 2009, 01:39 PM
Grillakis
Plato, I don't understand this:

http://www.mathhelpforum.com/math-he...03b4bbdf-1.gif and http://www.mathhelpforum.com/math-he...228e263f-1.gif.

I got the sequence of 2, 6, 10, etc. But i don't understand the above part of what you wrote?
• Mar 23rd 2009, 01:50 PM
Plato
Quote:

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

http://www.mathhelpforum.com/math-he...03b4bbdf-1.gif and http://www.mathhelpforum.com/math-he...228e263f-1.gif.

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?
• Mar 23rd 2009, 02:03 PM
Grillakis
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.
• Mar 23rd 2009, 02:47 PM
Plato
Look at this sequence: \$\displaystyle a_1 =0\$ then if \$\displaystyle n\ge2\$ define \$\displaystyle a_n = 2 - a_{n-1}\$.
Now you write out the next six terms. What do you see?