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anemone
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Let $U_1,\,U_2,\,\cdots$ be a sequence defined by $U_1=1$ and for $n>1$, $U_{n+1}=\sqrt{U_n^2-2U_n+3}+1$. Find $U_{513}$.
topsquark said:Well it's 33 isn't it?
Hmph. I got my count wrong. No more Excel for me!mathbalarka said:Close, but no cigar. $U_{510}$ is 33, not $U_{513}$,as 510 is 2 modulo 4, unlike 513.
(Lipssealed) I won't give out anything that will lead to a solution.
MarkFL said:My solution:
If we square the recursion, we may write:
\(\displaystyle \left(U_{n+1}-1\right)^2=\left(U_{n}-1\right)^2+2\)
Let:
\(\displaystyle V_n=\left(U_{n}-1\right)^2\)
And there results:
\(\displaystyle V_{n+1}=V_{n}+2\)
\(\displaystyle V_{n+2}=V_{n+1}+2\)
Subtracting the former from the latter, we obtain:
\(\displaystyle V_{n+2}=2V_{n+1}-V_{n}\)
We have the repeated characteristic root:
\(\displaystyle r=1\)
And so the closed form is:
\(\displaystyle V_n=c_1n+c_2\)
Using the initial conditions, we find:
\(\displaystyle V_1=c_1\cdot1+c_2=0\implies c_2=-c_1\)
\(\displaystyle V_2=c_1\cdot2+c_2=2\implies c_1=2,\,c_2=-2\)
Hence:
\(\displaystyle V_n=2(n-1)\)
Thus, we find:
\(\displaystyle V_{513}=2\cdot512=32^2\)
And so:
\(\displaystyle V_{513}=\left(U_{513}-1\right)^2\)
Since we must have \(\displaystyle 1\le U_n\forall n\in\mathbb{N}\), we take the positive root to obtain:
\(\displaystyle U_{513}=32+1=33\)
After I realized I had made an error in the initial conditions while I was away, it turns out Dan was correct after all.
The $U_{513}$ represents the 513th term of the sequence and is important in understanding the behavior and patterns of the sequence. It can also help in making predictions for future terms in the sequence.
To calculate the $U_{513}$, you need to first determine the recurrence relation for the sequence. Then, use the given initial terms of the sequence to find the pattern and use it to find the 513th term.
Sure, let's say we have a sequence defined by the recurrence relation $a_n = a_{n-1} + 3$, with the first term $a_1 = 5$. To find $U_{513}$, we first need to find the pattern of the sequence, which is adding 3 to the previous term. Then, we can use the formula $a_n = a_1 + (n-1)d$, where $d$ is the common difference, to find $U_{513}$. Plugging in the values, we get $U_{513} = 5 + (513-1)(3) = 1535$.
One common mistake is using the wrong recurrence relation or formula for the sequence. It is important to carefully read and understand the given information to ensure the correct calculation. Another mistake is not paying attention to the initial terms of the sequence, which can result in an incorrect pattern and final answer.
Yes, as long as the recurrence relation and the initial terms of the sequence are given, $U_{513}$ can be calculated. However, some sequences may have more complex patterns and require more steps to find the 513th term.