How to give inductive definition to sequence?

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Give inductive definition for the following:

1). 120, 60, 30, 15, 7.5, …
2). 4, 9, 19, 39, 79, …
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Kevin De Bruyne
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(Original post by TSRforum)
Give inductive definition for the following:

1). 120, 60, 30, 15, 7.5, …
2). 4, 9, 19, 39, 79, …
What two things do you need to define a sequence inductively?
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(Original post by SeanFM)
What two things do you need to define a sequence inductively?
No idea. I can only define arithmetic sequences using a and d.
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Kevin De Bruyne
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(Original post by TSRforum)
No idea. I can only define arithmetic sequences using a and d.
That's kind of like an inductive definition for a sequence, in that the first term, = a, and you're given that Image and
Image.

Does that help?
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(Original post by SeanFM)
That's kind of like an inductive definition for a sequence, in that the first term, = a, and you're given that a_1 = a and
a_{n+1} = a_{n} + d.

Does that help?
I don't think you can use that for the sequences I posted.
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Kevin De Bruyne
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(Original post by TSRforum)
I don't think you can use that for the sequences I posted.
Then the question would be impossible to solve

The answers aren't going to look exactly like that but something similar.

What is the first term in sequence 1? And what is the relationship between two consecutive terms?
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(Original post by SeanFM)
Then the question would be impossible to solve

The answers aren't going to look exactly like that but something similar.

What is the first term in sequence 1? And what is the relationship between two consecutive terms?
That's where the sequence starts and the relationship isn't stated.
There is no extra information, that's all that's given.
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Kevin De Bruyne
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(Original post by TSRforum)
That's where the sequence starts and the relationship isn't stated.
There is no extra information, that's all that's given.
You are reading my questions in the wrong way

That is where the sequence starts, yes, so the first term must be there at the start.

The relationship isn't stated but you should be able to identify it.
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(Original post by SeanFM)
You are reading my questions in the wrong way

That is where the sequence starts, yes, so the first term must be there at the start.

The relationship isn't stated but you should be able to identify it.
I know the relationship, but now how would I turn it into an inductive definition? Also here's another sequence: 1, 3, 11, 43, 171, … relationship is a bit harder to find there but I still can't turn it into a I.Definition.
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Kevin De Bruyne
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(Original post by TSRforum)
I know the relationship, but now how would I turn it into an inductive definition? Also here's another sequence: 1, 3, 11, 43, 171, … relationship is a bit harder to find there but I still can't turn it into a I.Definition.
Let's focus on the problems we have first of defining inductive definitions rather than moving onto a new problem.

If we had an arithmetic series 5, 15, 25, .... then you would tell me that a = 5 and d = 10. I'm hesitant to use this as an example as we are not to confuse the formula for the nth term, n = a + (n--1)d for an inductive definition because it is not. An inductive definition uses other terms to work out the n+1 th term.

So for that example, we can safely say that 5 is the first term in the sequence so a_{1} = 5 and the relationship between two terms is that you add 10 each time, so the definition is that
a_{n+1} = a_{n} + 10, so when n = 2 you use a1 to find that a2 = 15, use a2 to find a3 which is 25... and so on, which is the magic of induction.

So in Q1, I'll tell you that a1 = 120. What's the relationship between that and a2, and how can we express it like above?
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(Original post by TSRforum)
That's where the sequence starts and the relationship isn't stated.
There is no extra information, that's all that's given.
Sean isn't asking for extra information - you should be able to answer the questions by looking at the given values and *thinking*. Explicity: try to think what operation(s) you might use to get from 120 to 60, and from 60 to 30, and so on.
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(Original post by DFranklin)
Sean isn't asking for extra information - you should be able to answer the questions by looking at the given values and *thinking*. Explicity: try to think what operation(s) you might use to get from 120 to 60, and from 60 to 30, and so on.
I know how to find the relationship but then how do I turn it into a inductive definition?
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(Original post by TSRforum)
I know how to find the relationship but then how do I turn it into a inductive definition?
OK, explain what the relationship is.
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(Original post by DFranklin)
OK, explain what the relationship is.
x1/2 to get next term
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(Original post by TSRforum)
x1/2 to get next term
So what is an algebraic relationship connecting a given term u_n with the next term in the sequence, u_{n+1}
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(Original post by 16Characters....)
So what is an algebraic relationship connecting a given term u_n with the next term in the sequence, u_{n+1}
1/2Un?
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(Original post by TSRforum)
1/2Un?
Yes,  u_{n+1} = \frac{1}{2}u_n

Then this, together with your first term u_1 = 120 is your inductive definition:

u_1 = 120, u_{n+1} = \frac{1}{2}u_n
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(Original post by 16Characters....)
Yes,  u_{n+1} = \frac{1}{2}u_n

Then this, together with your first term u_1 = 120 is your inductive definition:

u_1 = 120, u_{n+1} = \frac{1}{2}u_n
Is this how it will always be written: Un+1 = Un + c or Un+1 = Un

Or can you have indices or brackets aswell?

e.g. Un+1 = (Un -3)^2 -3/4


If not then I just need to find out what I can times Un by then what I need to add or subtract after that to get the next term?
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(Original post by TSRforum)
Is this how it will always be written: Un+1 = Un + c or Un+1 = Un

Or can you have indices or brackets aswell?

e.g. Un+1 = (Un -3)^2 -3/4


If not then I just need to find out what I can times Un by then what I need to add or subtract after that to get the next term?
The relationship between successive terms could be anything, it could involve adding, multiplying or indices.

What it will actually involve in your exam depends on your specification. If it is for A Level I do not imagine the recurrence relationship (the relationship between two consecutive terms) will be too complicated.
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(Original post by 16Characters....)
The relationship between successive terms could be anything, it could involve adding, multiplying or indices.

What it will actually involve in your exam depends on your specification. If it is for A Level I do not imagine the recurrence relationship (the relationship between two consecutive terms) will be too complicated.
Edit: nvm
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