Question

Write a recursive formula for 16, 9, 7, 2, 5....

Answer 1

what is 16-9?

Answer 2

7

Answer 3

What's the term after 9 in the sequence?

Answer 4

What is your hypothesis?

Answer 5

9-2=7
an 7-2=5

Answer 6

Hypothesis?

Answer 7

ok but how write the formula?

Answer 8

name the terms in your head

Answer 9

what is the relationship between successive terms?

Answer 10

when you subtruct the first number by the second you will get the third and so on

Answer 11

exactly
the first two terms are just given
after that the following terms are generated according to the pattern you just identified

so call some number past the third number\[a_n\]how can we relate that number to the numbers preceding it\[a_{n-1},a_{n-2}\]?

Answer 12

an-7,an-2,an-5

Answer 13

I think you are misunderstanding the notation...

Answer 14

\[a_{n}\]is the nth number in the sequence\[16, 9, 7, 2, 5... \]\[a_1,a_2,a_3,a_4,a_5...\]so the number in the subscript is just telling you what term we are on

now, what is the relationship between\[a_5,a_4,\text{ and }a_3\]?

Answer 15

subtracting 1

Answer 16

write out a formula for \(a_5\) in terms of \(a_4\) and \(a_3\)

Answer 17

how do I write a formula

Answer 18

try it with number
let me make sure you understand what I am asking...
what is \(a_5\) ?

Answer 19

5

Answer 20

yes,
what is \(a_4\) ?
what is \(a_3\) ?
how are these three numbers ralated?

Answer 21

related*

Answer 22

4,3,2 they are decreasing as you go along by 1

Answer 23

no you have misunderstood\[a_1=16\]\[a_2=9\]\[a_3=7\]\[a_4=2\]\[a_5=5\]\[...\]the numbers in the bottom right (called subscripts) just tell you the term we are on

Answer 24

and will that be considered a formula

Answer 25

so in general when we write \[a_n=n^{th}\text{ term }\]to write a formula we need to generalize the relationship between any three numbers in the list, so no, listing out the given terms is how we start, not the answer

Answer 26

what is \(a_3\) ?

Answer 27

3

Answer 28

is it?
what is the third term on the list?

Answer 29

7

Answer 30

yes
what is \(a_2\) ?

Answer 31

9

Answer 32

yes
what is \(a_1\) ?

Answer 33

16

Answer 34

now you're getting the idea
now what is the relationship between those three numbers?

Answer 35

subtracting the first 2 to get the 3rd number

Answer 36

right
now write that out mathematically
(if you still can't do it with an then just write it out with the numbers....)

Answer 37

16-9=7,7-2=5

Answer 38

excellent
now all you need to do is rewrite this with the \(a_n\) notation and the formula comes out
instead of writing
16-9=7
write
a_?-a_?=a_?

fill the ?'s with the correct subscript (number of the term)

Answer 39

a-16=9=a-9=7=a-7=2=a-5

Answer 40

no, remember what \(a_1\) means
what is \(a_1\) ?

Answer 41

if you can't write this using the equation editor write it as
a_1=

Answer 42

16-a=9,9-a=7,7-a=2,a-2=5

Answer 43

what is a_1 ?

Answer 44

16

Answer 45

what is a_2 ?

Answer 46

9

Answer 47

what is a_3 ?

Answer 48

7

Answer 49

now you showed that
16-9=7
so rewrite the above expression with the a_n notation
a_?-a_?=a_?

(there should only be three terms)

Answer 50

a16-a9=a7

Answer 51

16 is not a16
16 is a_1

Answer 52

a16 would be the 16th term of the sequence (we don't know what that is yet)

you are over complicating the situation. all I want you to do is recognize the terms which you just did a moment ago

Answer 53

fill in the other two numbers
9=a_? (it's \(not\) a_9, it's the number of the \(term\))

Answer 54

a_1-a_2=7 a_3_a_4=2a_3-a_4=5

Answer 55

almost, but please don't write it all in a line like that, it is hard to read....
if you started writing
a_1-a_2=7
that is good, but you still need to sub for 7=a_?

Answer 56

what do you mean couldn't write like the way I wrote

Answer 57

did you try to write
a_1-a_2=7
a_3-a_4=2
a_3-a_4=5
???
if so that is not quite right, we want no numbers left
just call everything by the term name

Answer 58

so what is term is 7 (a_(what number??))
a_1-a_2=a_?

Answer 59

3 term

Answer 60

so we can write
a_1-a_2=a_3
understand??

Answer 61

yes

Answer 62

ok for the sake of simplicity let's turn this around
a_3=a_1-a_2
now what about the other terms?
a_4=?
a_5=?

Answer 63

a_4=a_2-a_3

Answer 64

nice!

Answer 65

a_5=a_3-a_4

Answer 66

ok, so now you need to find a way to write this in general for
a_n=?

Answer 67

whats n

Answer 68

to do this try calling n=5
what is the formula then?
\[a_5=a_3-a_4\]let \(n=5\)
\[a_n=a_?-a_?\]that is, how are the three subscripts related
(if n=5 then 3=n-?)

Answer 69

n is the unknown n'th term in the sequence

Answer 70

that is, whatever term n they want will specified by our formula

Answer 71

ok so how do i write the formula?

Answer 72

like I say, try it with any three numbers\[a_4=a_2-a_3\]if we rename \(4=n\) how would we write 3 in terms of n ?
(as 3=n-(something) )
i.e.
3=4-x