# Finding the simplest equation for the sequence: 1, 5, 21, 85... (Challenge for pros)

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Obviously, the number is going up by 4^(n-1) each time.

1st term) Let the first term = 1

2nd term) The term before + 4^1 = 5

3rd term) The term before + 4^2 = 21

4th term) The term before + 4^3 + 85

And so forth.

Without using trial and error, what is the easiest and best method to find a simplified equation for similar questions (e.g. sequences with differences going up by powers)?

I don't want the equation to rely on other terms, e.g. I don't want to include the "The term before +" bit. I want it to be a formula, so you can calculate the 800th term without calculating any other terms.

I am not doing this for education but to expand my maths, I just do not see a method how. I know the answer but not the method. I just don't like doing it via trial and error.

Thanks!

EDIT:

This is the answer. The challenge is to find the EASIEST and most reliable method of finding the equation of a sequence with differences of increasing powers, for instance, the one above is increasing by powers of 4. I have yet to find the easiest method, I have tried Google and messing about, yet my best method is simply trial and error and including (4^something) in the equation; however, I am certain there is a better method.

1st term) Let the first term = 1

2nd term) The term before + 4^1 = 5

3rd term) The term before + 4^2 = 21

4th term) The term before + 4^3 + 85

And so forth.

Without using trial and error, what is the easiest and best method to find a simplified equation for similar questions (e.g. sequences with differences going up by powers)?

I don't want the equation to rely on other terms, e.g. I don't want to include the "The term before +" bit. I want it to be a formula, so you can calculate the 800th term without calculating any other terms.

I am not doing this for education but to expand my maths, I just do not see a method how. I know the answer but not the method. I just don't like doing it via trial and error.

Thanks!

EDIT:

This is the answer. The challenge is to find the EASIEST and most reliable method of finding the equation of a sequence with differences of increasing powers, for instance, the one above is increasing by powers of 4. I have yet to find the easiest method, I have tried Google and messing about, yet my best method is simply trial and error and including (4^something) in the equation; however, I am certain there is a better method.

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#2

(Original post by

Obviously, the number is going up by 4^(n-1) each time.

1st term) Let the first term = 1

2nd term) The term before + 4^1 = 5

3rd term) The term before + 4^2 = 21

4th term) The term before + 4^3 + 85

And so forth.

Without using trial and error, what is the easiest and best method to find a simplified equation for similar questions (e.g. sequences with differences going up by powers)?

I don't want the equation to rely on other terms, e.g. I don't want to include the "The term before +" bit. I want it to be a formula, so you can calculate the 800th term without calculating any other terms.

I am not doing this for education but to expand my maths, I just do not see a method how. I know the answer but not the method. I just don't like doing it via trial and error.

Thanks!

**Konnichiwa**)Obviously, the number is going up by 4^(n-1) each time.

1st term) Let the first term = 1

2nd term) The term before + 4^1 = 5

3rd term) The term before + 4^2 = 21

4th term) The term before + 4^3 + 85

And so forth.

Without using trial and error, what is the easiest and best method to find a simplified equation for similar questions (e.g. sequences with differences going up by powers)?

I don't want the equation to rely on other terms, e.g. I don't want to include the "The term before +" bit. I want it to be a formula, so you can calculate the 800th term without calculating any other terms.

I am not doing this for education but to expand my maths, I just do not see a method how. I know the answer but not the method. I just don't like doing it via trial and error.

Thanks!

Still a term-to-term rule though

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#4

**Konnichiwa**)

Obviously, the number is going up by 4^(n-1) each time.

1st term) Let the first term = 1

2nd term) The term before + 4^1 = 5

3rd term) The term before + 4^2 = 21

4th term) The term before + 4^3 + 85

And so forth.

Without using trial and error, what is the easiest and best method to find a simplified equation for similar questions (e.g. sequences with differences going up by powers)?

I don't want the equation to rely on other terms, e.g. I don't want to include the "The term before +" bit. I want it to be a formula, so you can calculate the 800th term without calculating any other terms.

I am not doing this for education but to expand my maths, I just do not see a method how. I know the answer but not the method. I just don't like doing it via trial and error.

Thanks!

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(Original post by

Odd choice of rule I would have gone with

Still a term-to-term rule though

**TenOfThem**)Odd choice of rule I would have gone with

Still a term-to-term rule though

There is an answer, and I am wondering if there is a method that people use to find expressions (without relying on other terms) of a sequence with differences going up in powers/exponentially.

This is what troubled me.

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(Original post by

Just a question, what year are you in?

**#Unknown**)Just a question, what year are you in?

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#8

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#9

(Original post by

Yes, but it can be expressed without relying on other terms.

There is an answer,

**Konnichiwa**)Yes, but it can be expressed without relying on other terms.

There is an answer,

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#10

It's not always minus the first term. I don't really have a set method. Just practice it a bit.

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(Original post by

I'd like to know the answer to this tbh. I'm not the best of coming up with equations for series. DO you have a method?

**keromedic**)I'd like to know the answer to this tbh. I'm not the best of coming up with equations for series. DO you have a method?

It should be a nice challenge finding the method.

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#12

**keromedic**)

I'd like to know the answer to this tbh. I'm not the best of coming up with equations for series. DO you have a method?

I should have left well alone with the OPs original formula

Then think about summing as it is just a GP being summed

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(Original post by

It's always the common difference, minus the first number, divided by some ratio. The common difference in this series is

**keromedic**)It's always the common difference, minus the first number, divided by some ratio. The common difference in this series is

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#14

(Original post by

In this question - easy

I should have left well alone with the OPs original formula

Then think about summing as it is just a GP being summed

**TenOfThem**)In this question - easy

I should have left well alone with the OPs original formula

Then think about summing as it is just a GP being summed

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#15

(Original post by

That's the way you are supposed to do it, but instead of the term before, you use u(n) [nth term]=u(n-1)[the term before] + 4^(n-1), with u(1)[first term] being 1, does this clear things up for you?

BTW defining a sequence like this is called a RECURRENCE RELATION I think:-)

Posted from TSR Mobile

**majmuh24**)That's the way you are supposed to do it, but instead of the term before, you use u(n) [nth term]=u(n-1)[the term before] + 4^(n-1), with u(1)[first term] being 1, does this clear things up for you?

BTW defining a sequence like this is called a RECURRENCE RELATION I think:-)

Posted from TSR Mobile

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#17

That is a different sequence to the one the rest of us have found

But, of course, with only a finite number of terms there will be different rules

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#20

(Original post by

How do you come up with them? Seems like a very impressive skill IMO!

**krisshP**)How do you come up with them? Seems like a very impressive skill IMO!

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