Question

How can you represent the terms of a sequence explicitly? How can you represent them recursively?

Provide mathematical examples to support your opinions.

Answer 1

If a sequence has the numbers 1, 5, 9, 13, 17.
Explicitly a^n= 4n+5
Recursively a^n = a^(n-1)+4

Answer 2

Is my answer correct?

Answer 3

not quite, do you propose to start with n=1 or n=0 ?

Answer 4

Shouldn't it start with n=1? (I just started studying this)

Answer 5

you can do it either way
but look what happens in your explicit formula when n=1. what is the value?

Answer 6

9?

Answer 7

That wouldn't work out if so..

Answer 8

right, so how can we get 5?
we would have to start with n=0
like I say, that's ok, but then how can we get the 1?

Answer 9

I'm not sure how I would go about that, I would have to write a new explicit formula right?

Answer 10

yeah, but it won't be too different
consider the first terms written differently (let's start from n=0)\[a_0=1\\a_1=1+4\\a_2=1+4+4\\~~~~~\vdots\]any ideas yet?

Answer 11

a^n=1+4^n

Answer 12

well ^ usually means exponent...

Answer 13

do you mean\[a_n=1+4n\]

Answer 14

Yes sorry. That should've been written like a_n=1+4^n right?

Answer 15

again, 4^n is exponent
do you mean 4 to the power of n?

Answer 16

Yes, I thought that was how you typed it?

Answer 17

it is, but the answer should not be an exponent
we are adding 4 every time. exponent would be multiplying by 4 every time

Answer 18

Ohhhhh! I just realized what I was doing. \[a _{n}=1+4n\]

Answer 19

yes, much better :)

Answer 20

Okay, so how would we figure out the recursive equation?

Answer 21

ok so your recursive is almost right, except again you have no way of getting to the first term.

however to do this for recursive functions, the method is to simply define the first term, which if you think about it is necessary. so what is the value of \(a_0\) ?

Answer 22

In my equation the value of my first term would be 3.... But I need to find a way to write it so it is 1.

Answer 23

oh, ok look you are writing exponents all over the place. there should be no exponents.
what you want instead of exponents on the left are subscripts, which you can write with your keyboard as
a_n=a_(n-1)+4
(I know it looks ugly, you can use the equation editor or LaTeX if you want to make it like mine)

Answer 24

you are writing\[a^n=a^{n-1}+4\]it should be\[a_n=a_{n-1}+4\]

Answer 25

Okay.. I wasn't sure how I was supposed to write subscript.

Answer 26

well how did you get your first value as 3 ?

Answer 27

by plugging it into my old recursive equation.

Answer 28

Not 3... sorry that should be 4.

Answer 29

what did you plug in? how did you get the value of \(a_{n-1}\)?

Answer 30

Shouldn't I plug in 1?

Answer 31

for n? then you get\[a_1=a_0+4\]so you still have the problem of how to get \(a_0\)

Answer 32

you just plugged in \(0\) for \(a_0\), which is unjustified

Answer 33

Okay so I'm starting with my first term (a_0) and I'm plugging in 1 for that right?

Answer 34

yes, so we need to state that in our formula
recursive formulas always need a base case like this, otherwise they can never get started

Answer 35

so to make our recursive formula complete we need to actually say\[a_0=1\]in addition to what you wrote

Answer 36

does that make any kind of sense? that without a base case where we define one term to be a particular value, our recursive formula won't work?

Answer 37

Yeah that makes sense.

Answer 38

Well I hope that helped
See you around!

Answer 39

Wait what was my recursive formula going to be?

Answer 40

\[a_0=1\\a_n=a_{n-1}+4\]