MHB Finding $k_{513}$ in the Sequence $k_1,k_2,\cdots$

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The sequence defined by \( k_1 = 1 \) and \( k_{n+1} = \sqrt{k_n^2 - 2k_n + 3} + 1 \) is discussed with the goal of finding \( k_{513} \). Participants engage in solving the problem, with some solutions being deemed incorrect while others are praised. MarkFL and anemone provide correct solutions, while kaliprasad initially faces criticism but later receives acknowledgment for their accurate approach. The conversation highlights collaborative problem-solving and verification of mathematical solutions. Ultimately, the focus remains on determining the value of \( k_{513} \).
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Let $k_1,k_2,\cdots$ be a sequence defined by $k_1=1$ and for $n \ge 1$, $k_{n+1}=\sqrt{k_n^2-2k_n+3}+1$. Find $k_{513}$.
 
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anemone said:
Let $k_1,k_2,\cdots$ be a sequence defined by $k_1=1$ and for $n \ge 1$, $k_{n+1}=\sqrt{k_n^2-2k_n+3}+1$. Find $k_{513}$.

32

solved as

Take 1 to left and square both sides

(K(n+1) – 1)^2 = k(n)^2 – 2k(n) + 3 = (k(n)-1)^2 + 2
Or (K(n+1) – 1)^2 = (k(n)-1)^2 + 2
(k(2)-1)^2 = (k(1)-1)^2 + 2
(k(3)-1)^2 = (k(2)-1)^2 + 2 = = (k(1)-1)^2 + 2 * 2
Proceeding we see hat = (k(n)-1)^2 = = (k(1)-1)^2 + 2(n-1)
One can prove it by induction
Put n = 513 to get k(513) = (k(1)-1)^2 + 2 * 512 = 1024
K(513) = 32 as it has to be positive
 
kaliprasad said:
32

solved as

Take 1 to left and square both sides

(K(n+1) – 1)^2 = k(n)^2 – 2k(n) + 3 = (k(n)-1)^2 + 2
Or (K(n+1) – 1)^2 = (k(n)-1)^2 + 2
(k(2)-1)^2 = (k(1)-1)^2 + 2
(k(3)-1)^2 = (k(2)-1)^2 + 2 = = (k(1)-1)^2 + 2 * 2
Proceeding we see hat = (k(n)-1)^2 = = (k(1)-1)^2 + 2(n-1)
One can prove it by induction
Put n = 513 to get k(513) = (k(1)-1)^2 + 2 * 512 = 1024
K(513) = 32 as it has to be positive

Thanks for participating kaliprasad...but your answer isn't correct. I'm sorry.:(
 
Here is my solution:

We are given the recursive algorithm:

$$k_{n+1}=\sqrt{k_n^2-2k_n+3}+1$$ where $$k_1=1$$

If we subtract 1 from both sides and square, we obtain:

$$\left(k_{n+1}-1 \right)^2=\left(k_{n}-1 \right)^2+2$$

If we define:

$$U_n=\left(k_{n}-1 \right)^2$$

we then obtain the linear difference equation:

$$U_{n+1}-U_{n}=2$$ where $$U_1=0$$

The homogeneous solution is:

$$h_n=c_1$$

and the particular solution is:

$$p_n=c_2n$$

Substituting the particular solution into the difference equation, we find:

$$c_2(n+1)-c_2n=2\implies c_2=2$$

Thus, the general solution is:

$$U_n=c_1+2n$$

We may now use the initial value to determine the parameter $c_1$:

$$U_1=c_1+2=0\implies c_1=-2$$

And so the solution satisfying the given conditions is:

$$U_n=-2+2n=2(n-1)$$

Hence, we find:

$$U_{513}=2(513-1)=1024$$

Thus:

$$k_{513}=\sqrt{U_{513}}+1=33$$
 
MarkFL said:
Here is my solution:

We are given the recursive algorithm:

$$k_{n+1}=\sqrt{k_n^2-2k_n+3}+1$$ where $$k_1=1$$

If we subtract 1 from both sides and square, we obtain:

$$\left(k_{n+1}-1 \right)^2=\left(k_{n}-1 \right)^2+2$$

If we define:

$$U_n=\left(k_{n}-1 \right)^2$$

we then obtain the linear difference equation:

$$U_{n+1}-U_{n}=2$$ where $$U_1=0$$

The homogeneous solution is:

$$h_n=c_1$$

and the particular solution is:

$$p_n=c_2n$$

Substituting the particular solution into the difference equation, we find:

$$c_2(n+1)-c_2n=2\implies c_2=2$$

Thus, the general solution is:

$$U_n=c_1+2n$$

We may now use the initial value to determine the parameter $c_1$:

$$U_1=c_1+2=0\implies c_1=-2$$

And so the solution satisfying the given conditions is:

$$U_n=-2+2n=2(n-1)$$

Hence, we find:

$$U_{513}=2(513-1)=1024$$

Thus:

$$k_{513}=\sqrt{U_{513}}+1=33$$

Bravo, MarkFL:cool: and thanks for participating!
 
anemone said:
Thanks for participating kaliprasad...but your answer isn't correct. I'm sorry.:(

MarkFL and anemone
thanks

my solution
Proceeding we see hat = (k(n)-1)^2 = = (k(1)-1)^2 + 2(n-1)
I had done correct till the aboveI forgot to take 1 to the right
(k(513) - 1)^2 = 1024

or k(513) = 33

so approach was right but not taking 1 to the right was an oversight
 
Last edited:
kaliprasad said:
MarkFL and anemone
thanks

my solution
Proceeding we see hat = (k(n)-1)^2 = = (k(1)-1)^2 + 2(n-1)
I had done correct till the aboveI forgot to take 1 to the right
k(513) - 1 = 1024

or k(513) = 33

so approach was right but not taking 1 to the right was an oversight

I am sorry kaliprasad...I checked your approach but couldn't locate the mistake (the honest kind, of course) and now everything seems perfect about your solution! Well done, kaliprasad!
 
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