Question

i don't get recursive or explicit formulas at all! nEED hELP!

Answer 1

What is it that you dont get. Tell us how far you got.

Answer 2

http://www.ck12.org/book/CK-12-Math-Analysis/r26/section/7.1/

Answer 3

the formulas confuse me and i need to know it by friday!

Answer 4

Read this :

Vaguely, a recursive formula has the
property of referencing itself. In particular, the following formula
is recursive:

F[n] = F[n-1]

This means that if you want to know what F[5] is, you plug in 5 for
n to get F[5-1] = F[4]. But that's not much of an answer, is it?
Well, what's F[4]? We use the above formula again and we get
F[4] = F[4-1] = F[3]. In fact, we see that:

F[n] = F[n-1] = F[n-2] = F[n-3] = ...

While we don't know what all of these values are, the recursive
formula tells us that they are all the same. But is it true that
F[n] = F[n-1/2]? Not necessarily. For example, let's add two
conditions:

F[0] = 0, F[1/2] = 1

Then clearly, F[n] = 0 for all integers n, and F[n+1/2] = 1 for all
integers n. Also, it is important to note that simply because we
have defined F[0] and F[1/2], F[n] for other values of n is not
necessarily determined from this and its recursive formula. For
example, what is F[1/4]? F[2/3]? F[3.14159]? We simply don't know
from the information given thus far. However, say we defined F[n] to
be:
F[n] = F[n-1]
F[x] = x for all real x greater or equal to 0 and less than 1

(We can write the second line as "F[x] = x for x in [0,1)" - the
[0,1) is an interval which includes all numbers greater than or equal
to 0 but less than 1.) Then the graph of F[n] looks like this:

1_|
/| / / / /
/ | / / / /
______________________/__|/___/___/___/______________ n
-1 0| 1 2 3
|
|

It is composed of a series of 45-degree angle segments. Note that
F[1] = 0, but F[0.99999] = 0.99999; in fact, for a number b
"arbitrarily close to" 1 but less than it, F[b] = b.

Anyway, I digress. This isn't about limits or weird functions, but
the difference between recursive and explicit formulas. To this end,
I will give you the explicit formula for the same function F[n],
defined for all real values n. It is

G[n] = n - Floor[n],

where Floor[n] is the greatest integer less than or equal to n. So if
n = -1, G[-1] = -1 - Floor[-1] = 0. Similarly:

G[-3.5] = -3.5 - Floor[-3.5] = -3.5 - (-4) = 0.5

If we calculate this using the recursive formula:

F[-3.5] = F[-2.5] = F[-1.5] = F[-0.5] = F[0.5] = 0.5

So they are the same!

Notice, then, that the two definitions:

F[n] = F[n-1], F[x] = x for x in [0,1)

and

G[n] = n - Floor[n],

are quite different, not only in notation but in how their values are
calculated. The first (recursive) relies on some set of initial
conditions (F[x] = x for x in [0,1)) and a relation which allows you
to travel this "path" to the initial condition. The second is very
straightforward, and does not rely on any information other than how
to manipulate the input value to get the value of the function. In
particular, it "does not rely on itself."

Another example, on nonnegative integers n, is:

Recursive: F[n] = n*F[n-1], F[0] = 1
Explicit: G[n] = n!

They are the same function, the factorial function, and I will leave
it as an exercise for you to determine why.

A third example, also defined on nonnegative integers n:

Recursive: F[n] = F[n-1] + F[n-2], F[0] = 0, F[1] = 1
Explicit: G[n] = (((1+Sqrt[5])/2)^n - ((1-Sqrt[5]))^n)/Sqrt[5]

These are also the same function, this time for the Fibonacci numbers,
0, 1, 1, 2, 3, 5, 8, 13, .... To show that these are indeed the same
is quite easy; one need only verify that G[0] = 0, G[1] = 1, and
G[n] = G[n-1] + G[n-2], just like F. This I also leave as an
excercise.

I hope that this gives you an idea of what the difference between a
recursive and explicit formula is. The former usually "refers" to
itself -- notice that in each case, the function F appears on both
sides of the equation, albeit for different values. The latter,
however, does not in general have this property. Note however that

G[n] = G[0] + n*G[1]

is not a recursive function, because G[0] and G[1] are constants which
do not depend on n.

Answer 5

thank you @tanya123

Answer 6

@tanya123 i still don't get it?

Answer 7

@jdoe0001

Answer 8

@satellite73 please help me!

Answer 9

@ShortyB

Answer 10

@zepdrix

Answer 11

@Hero @iPwnBunnies

Answer 12

Please anyone!

Answer 13

Recursive means each term after the first is defined by the term right before it. You have to state the first term, then a rule. Explicit means just state a rule that works if you plug in the number of the term you want, rather than an actual term.
for example 41,46,51,56,61,...

So, since 5 is added to each term to get the next,
recursive: a1 = 41, an = a(n - 1) + 5
explicit: an = 5n + 36

Then since each term is ten times the previous,
recursive: a1 = 1; an = 10 • a(n - 1)
explicit: an = 10 ^ (n - 1)

Then since each term is multiplied by 2,
recursive: a1 = 3, an = 2 • a(n - 1)
explicit: an = 3 • 2 ^ (n - 1)

Answer 14

i got this from Google tbh lol

Answer 15

explicit: an = 5n + 36 ?

Answer 16

find a10.

Answer 17

i guess they did that because they subtracted

Answer 18

im really sorry i cant help

Answer 19

@lacrosseplayer22 can you help

Answer 20

i don't get it still!

Answer 21

hold on a sec.

Answer 22

ok so do you know that recursive means?

Answer 23

ok im having internet trouble if my profile pic. keeps on flashing its not my fault. cant control it.

Answer 24

sorry im confused. I was reading the ck12 link. what part of it don't u get? u must understand something @baby456 explain to me what u know about recursive and explicit formulas

Answer 25

its a sequence.

Answer 26

i"m seriously confused!

Answer 27

what formula confuses you? what is your question?

Answer 28

just saying I dont get it, when there is 500+ words explaining it does not help. Exactly what do you not "get"?

Answer 29

each time we get our answer by using our last answer.....

Answer 30

one example

\(f(1) = 1\\
f(n)=f(n-1)+1\)

so

\(f(1) = 1, \\f(2) = f(2-1)+1 = f(1)+1 = 1+1 = 2\\f(3) = f(3-1) +1 = f(2) + 1 = 2+1 = 3\)

Answer 31

let say that we have the following recursive definition:
a(1)=1
a(n+1)=2a(n)+3, for n=1, 2, 3...

Note that a recursive definition has 2 parts - the initial condition, in this case a(1)=1
and a recursive part, in this case a(n+1)=a(n)
It also tells you the range of n for which the recursive part is valid, in this case n=1,2,3...

Answer 32

ok I let

Answer 33

@zzz0ck3r continue

Answer 34

I think we have explained it enough, the OP can ask a question if he/she is confused by something

Answer 35

i guess i get it little more.

Answer 36

concentrate on the recursive part first. The explicit formula canbe discussed after you have understood the recursive part.