# Find Formula for nth Term in Tuff Sequence

• MHB
• Wilmer
In summary, to find the formula for the nth term in a tuff sequence, you will need to identify the pattern in the sequence and use algebraic equations. There is no specific method for finding the formula, but common techniques include identifying differences, using equations, and creating a table. It is not recommended to solely rely on a calculator. If struggling, seek assistance. To check if your formula is correct, you can plug in values for n or use a calculator.
Wilmer
This sequence is similar to Sloane's #A010551; one further multiplication is made...
Code:
n= 0  1  2  3  4  5  6   7   8    9   10    11     12
1  1  1  1  2  4  8  24  72  216  864  3456  13824...
Find a formula for the nth term.

Wilmer said:
This sequence is similar to Sloane's #A010551; one further multiplication is made...
Code:
n= 0  1  2  3  4  5  6   7   8    9   10    11     12
1  1  1  1  2  4  8  24  72  216  864  3456  13824...
Find a formula for the nth term.

This sequence is generated by:
$$a_n = \tau_n a_{n - 1} ~ \text{with} ~ a_0 = 1$$
Where $\tau_n$ is given by:
$$\tau_n = \begin{cases} 1 ~ ~ &\text{if} ~ 1 \leq n \leq 3 \\ 2 ~ ~ &\text{if} ~ 4 \leq n \leq 6 \\ 3 ~ ~ &\text{if} ~ 7 \leq n \leq 9 \\ &\cdots \end{cases}$$
It's easy to see that $\tau_{3n} = n$ and since every three elements 1, 2, 3, then 4, 5, 6, then 7, 8, 9, etc. are the same, we have $\tau_{3n - 2} = \tau_{3n - 1} = n$ and $\tau_{3n + 1} = \tau_{3n + 2} = n + 1$.
Now observe that:
$$a_{3n} = \tau_{3n} a_{3n - 1} = \tau_{3n} \tau_{3n - 1} a_{3n - 2} = \tau_{3n} \tau_{3n - 1} \tau_{3n - 2} a_{3(n - 1)} = n^3 a_{3(n - 1)}$$
So that by induction we have:
$$a_{3n} = (n!)^3$$
Then:
$$a_{3n + 1} = \tau_{3n + 1} a_{3n} = (n + 1) (n!)^3$$
And:
$$a_{3n + 2} = \tau_{3n + 2} a_{3n + 1} = \tau_{3n + 2} \tau_{3n + 1} a_{3n} = (n + 1)^2 (n!)^3$$
And so we can conclude:
$$a_n = \begin{cases} (m!)^3 ~ ~ &\text{if} ~ n = 3m \\ (m + 1) (m!)^3 ~ ~ &\text{if} ~ n = 3m + 1 \\ (m + 1)^2 (m!)^3 ~ ~ &\text{if} ~ n = 3m + 2 \end{cases}$$
Or, more directly:
$$a_n = \left ( \left \lfloor \frac{n}{3} \right \rfloor + 1 \right )^{n \bmod{3}} \left ( \left \lfloor \frac{n}{3} \right \rfloor ! \right)^3$$
I suppose this can be easily generalized to any similar $\tau_n$ sequence of the form 1, 1, $k$ times, 2, 2, $k$ times, etc.. but I haven't tried.​

Nice, Bac! Mine's simpler:

nth term = a * b * c, where:
a = [FLOOR(n/3)]!
b = [FLOOR((n+1)/3)]!
c = [FLOOR((n+2)/3)]!

Wilmer said:
Nice, Bac! Mine's simpler:

nth term = a * b * c, where:
a = [FLOOR(n/3)]!
b = [FLOOR((n+1)/3)]!
c = [FLOOR((n+2)/3)]!

Ah, yep, that works too. Not bad :)

Hi,

You ask for one formula, I bring you a non countable set of formulas

Let $c=(1,1,1,1,2,4,8,24,72,216,864,3456,13824,r)^{T}\in \Bbb{R}^{14}$ with $r\in \Bbb{R}$.

Define $A=\left(\begin{array}{ccccc} 1 & 0 & 0 & \ldots & 0\\ 1^{0}& 1 & 1^{2} & \ldots & 1^{13}\\ 2^{0}& 2 & 2^{2} & \ldots & 2^{13}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 13^{0} & 13 & 13^{2}& \ldots & 13^{13}\end{array}\right)$

Call $b=(b_{0}, b_{1},\ldots , b_{13})^{T}\in \Bbb{R}^{14}$, the solution of the linear equations

$Ab=c$

And now consider the polynomial $p_{r}(x)=\displaystyle\sum_{i=0}^{13}b_{i}x^{i}$.

Then $p_{r}(0)=1, p_{r}(1)=1, \ldots , p_{r}(12)=13824, p_{r}(13)=r$.

And works for every $r\in \Bbb{R}$

The n-th term will be now $p_{r}(n)$

## 1. How do I find the formula for the nth term in a tuff sequence?

To find the formula for the nth term in a tuff sequence, you will need to first identify the pattern in the sequence. Look for any repeating numbers or operations between each term. Once you have identified the pattern, you can use algebraic equations to find the formula for the nth term.

## 2. Is there a specific method for finding the formula for the nth term in a tuff sequence?

There is no specific method for finding the formula for the nth term in a tuff sequence. However, some common techniques include identifying the differences between each term, using algebraic equations, or creating a table of values to help find the pattern.

## 3. Can I use a calculator to find the formula for the nth term in a tuff sequence?

While a calculator can be useful for performing calculations and checking your work, it is not recommended to solely rely on a calculator to find the formula for the nth term in a tuff sequence. It is important to understand the underlying concepts and patterns in the sequence.

## 4. What should I do if I am unable to find the formula for the nth term in a tuff sequence?

If you are struggling to find the formula for the nth term in a tuff sequence, it may be helpful to seek assistance from a tutor or teacher. Sometimes having a second set of eyes can help identify the pattern or provide helpful tips for solving the problem.

## 5. How can I check if my formula for the nth term in a tuff sequence is correct?

One way to check if your formula for the nth term in a tuff sequence is correct is to plug in different values for n and see if the resulting terms match the original sequence. You can also use a calculator to check if your formula produces the correct outputs for each term.

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