Sub Scripts

[Holoob]

Hi thanks for looking at this problem.

If the Sequence {a_n} is defined by {a_0}=1 and (a_n+1)=(2a_n)+(2) for n=0,1,2.... What is {a_3}?

Note: Underscore mean subscript possibly a log not sure.

This came straight out of the College board so I'm sure anyone who got a 700 or better can do this problem.

Not sure how but the answer is 22.

Good Luck! Ganbarimashyo:mrgreen:

 

[Quaid]

(a_n+1)

The typing above means a_n + 1


What you want is a_n+1


So, type it like a_[n+1]

(2a_n)+(2)

The parentheses above are unnecessary, but that could be shown as

2(a_n) + 2

a_0 = 1

a_[n+1] = 2(a_n) + 2, for n = 0,1,2...

What is a_3 ?

Not sure how but the answer is 22.

This is what we call a recursive sequence; this means that the next number in the sequence depends upon the previous number.

In other words, the symbol a_[n+1] represents the number that comes after a_n in the sequence.


To calculate a_3, you need the value for a_2

To calculate a_2, you need the value for a_1

To calculate a_1, you use the given value for a_0


I'll start for you:

Let n = 0

Then a_[n+1] = a_1

a_1 = 2(a_0) + 2

a_1 = 2(1) + 2

a_1 = 4


Now, you continue for n = 1 and find a_2 by using a_1 = 4

Then, use that result to see why a_3 = 22


If you get stuck, please show your work so far. Cheers :cool:

 

[Dale10101]

A thought

Hi thanks for looking at this problem.

If the Sequence {a_n} is defined by {a_0}=1 and (a_n+1)=(2a_n)+(2) for n=0,1,2.... What is {a_3}?

Note: Underscore mean subscript possibly a log not sure.

This came straight out of the College board so I'm sure anyone who got a 700 or better can do this problem.

Not sure how but the answer is 22.

Good Luck! Ganbarimashyo:mrgreen:

Am not a mentor but I might be closer to your state of confusion.

I find it easier initially forget about the indexes and focus on ... I am given a first element of a sequence and the ability to calculate the next element in terms of the preceding element, which is exactly what a_n+1 means and why it is given. It is simple to advance up the sequence just writing down each next result and forget about the indexes .... the only question is when to stop? ... at:

(n,f(n)) = (0,f(0)), (n,f(n)) = (1,f(1)), (n,f(n)) = (2,f(2)) ... plausible because the number of results is n = 3 f(0), f(1), f(2), i.e. is that what n refers to?

or

(n,f(n)) = (0,f(0)), (n,f(n)) = (1,f(1)), (n,f(n)) = (2,f(2)), (n,f(n)) = (3,f(3))... plausible because the name of the last result by its index is 3. , i.e. is that what n refers to?

Since you know that a_3 = 22, what n means is or will be clear.

The essential meaning of a recursive definition is pretty simple, it is the indexing system that always seemed confusing to me. For example when is 3 = 4 (informally speaking) ? When the 3 refers to the name of the fourth element of a counting sequence beginning with zero .... 0,1,2,3. Well, I hope I am making some sense for you.

 

[Holoob]

thanks that was all too much

<sub>I should have been more clear.

The sequence {A_n} is defined by {A_0=1} and {A_(n+1)=2a_n+2} for n=0,1,2,3.

What is {A_3}?

The possible answers are....

8

10

16

20

22

Thanks</sub>

 

[Quaid]

that was all too much

Okay -- thanks for letting us know this!

(As you did not explain what you already understand about sequences or this exercise, we guessed at what information might help you begin.)

Let's try again!

For this attempt, I'm not going to use symbol n or any subscripts; I'll use a different set of symbols for the elements of the sequence. Hopefully, this will help.

Here is the sequence notation: {a, b, c, d, e, f, ...}

Clearly, symbol a represents the first element in the sequence, symbol b represents the second element, and so on.


Here is the pattern of your recursive sequence:

a = 1

b = 2a + 2

c = 2b + 2

d = 2c + 2

e = 2d + 2

f = 2e + 2

and so on ...

Do you understand this recursive pattern, as I've presented it?


The FIRST element's value is given to you: a = 1

The value of every element AFTER the first one is found by doubling the previous element, followed by adding 2.


Let's calculate the second element's value:

b = 2a + 2


We already know the value of a, so we replace symbol a with its value:

b = 2(1) + 2


Simplify:

b = 4


Now that we know the value of the second element, we can find the next element's value:

c = 2b + 2


Substitute 4 for b:

c = 2(4) + 2


Simplify:

c = 10



Is this enough information for you to confirm that the value of the fourth element is 22?

If not, please show your work OR ask a specific question.

Cheers! :cool:


PS: Once you understand the recursive pattern AND how to calculate element values, then we can go back to discussing the subscript notation.


NOTE: In my first reply, I typed an incorrect name. (I wrote "regressive" instead of "recursive".) Sorry about that mistake.

 

[Dale10101]

Well

Sure you are...nost of us are helpers/mentors...pay is atrocious

Well ... helper maybe, but say ... are you sure this is not a sneaky way of having somebody to talk to on those occasional occasions when you err and are sent to the corner?