# Finding formula for nth term in sequence

• I
• JimbleJambler
In summary, the conversation discusses finding a formula for the nth term, which involves the function f_n being defined in terms of itself and previous values of the index. The sum in the formula ends with n, but the "mod 1" can be ignored and applied afterwards. It is suggested to work with integers mod 10 and that the recurrence is similar to a shift register random number generator.
JimbleJambler
##\displaystyle f_{n} = ((\sum_{k= \lfloor{\frac{n}{2}}\rfloor}^{n-1}f_{k}) mod 1) + 0.1##
##\displaystyle f_{1} = 0##

I really would like to know where to begin for finding a formula for the nth term, I wrote out a bunch of the terms and couldn't really eyeball a pattern of any sort. I noticed sometimes it seems to be that f(n) = (f(n-1) * 2) mod 1, but that's not very consistent.

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Clarify your definition. You have $$f_n$$ defined in terms of itself as well as previous values of the index.

Are you sure the sum ends with n? You use fn in the definition of fn.
You can ignore the "mod 1" and apply it afterwards in a suitable way, that won't change the sum.

mathman said:
Clarify your definition. You have $$f_n$$ defined in terms of itself as well as previous values of the index.
mfb said:
Are you sure the sum ends with n? You use fn in the definition of fn.
You can ignore the "mod 1" and apply it afterwards in a suitable way, that won't change the sum.
Sorry I fixed it!

It's easier to work with integers mod 10. I can only simplify it a bit.
Since f(2k) = f(k)+ ... + f(2k-1) + 1,
f(2k+1) = f(k)+ ... + f(2k-1) +f(2k) +1 = 2f(2k) so the odd values aren't interesting
f(2k+2) = 4f(2k) - f(k)
The recurrence is rather like a shift register random number generator, only with a growing shift register.
https://en.wikipedia.org/wiki/Linear-feedback_shift_register

## What is a sequence?

A sequence is a set of numbers that follow a specific pattern or rule. Each number in a sequence is called a term.

## What is the nth term in a sequence?

The nth term in a sequence refers to the general term or formula that can be used to find any term in the sequence, given its position or index.

## How do you find the formula for the nth term in a sequence?

The formula for the nth term in a sequence can be found by analyzing the pattern or rule of the sequence. This can involve looking for common differences or ratios between terms, using algebraic equations, or using other mathematical techniques.

## Why is it important to find the formula for the nth term in a sequence?

Finding the formula for the nth term in a sequence allows us to predict and calculate any term in the sequence without having to manually list out all the terms. This is particularly useful when dealing with large or complex sequences.

## Are there any shortcuts for finding the formula for the nth term in a sequence?

Yes, there are some common patterns and techniques that can be used to quickly find the formula for the nth term in a sequence. These include arithmetic and geometric sequences, as well as using the sum and product of consecutive terms.

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