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

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!

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!

Odd choice of rule I would have gone with $U_{n+1} = 4U_n + 1$

Still a term-to-term rule though

Like in this case $U_n=\dfrac{4^n-1}{3}$?

Like in this case $U_n=\dfrac{4^{n+1}-1}{3}$?

Just a question, what year are you in?

Duuuuuuurrrrrrrrr me

Good spot

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

There is an answer,

Yes, that is the answer but I cannot find the method.


13.

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'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's always the common difference, minus the first number, divided by some ratio. The common difference in this series is $4^n$

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

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:-)

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Yes, that is the answer but I cannot find the method.


13.

It can also be written as a cubic equation.
$u_n=6n^3-30n^2+52n-27$

Like in this case $U_n=\dfrac{4^n-1}{3}$?

It can also be written as a cubic equation.
$u_n=6n^3-30n^2+52n-27$

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