Solved on Mar 07, 2024

Redefine the Fibonacci sequence starting with $X_0=2, X_1=3$ or $X_0=3, X_1=5$ . Find the correct solution when the sequence is indexed 5 steps ahead.

STEP 1

Assumptions
1. The sequence follows the difference equation $X_{n+2} = X_{n+1} + X_{n}$ .
2. The original sequence starts with $X_{0} = 1$ and $X_{1} = 2$ .
3. We need to find the new constants for the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ , and again for $X_{0} = 3$ and $X_{1} = 5$ .
4. We are looking for a pattern in the solutions when re-indexing the sequence.
5. We need to determine the correct solution after indexing 5 steps ahead, effectively making $n=5$ the new $n=0$ .

STEP 2

First, let's find the solution for the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ .
We know that $X_{2} = X_{1} + X_{0}$ , so:
$X_{2} = 3 + 2 = 5$

STEP 3

Now, find $X_{3}$ using the difference equation:
$X_{3} = X_{2} + X_{1} = 5 + 3 = 8$

STEP 4

Next, find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 8 + 5 = 13$

STEP 5

We continue this process to find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 13 + 8 = 21$

STEP 6

Now we have the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ as $\{2, 3, 5, 8, 13, 21, \ldots\}$ .

STEP 7

Next, let's find the solution for the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ .
Again, we use the difference equation to find $X_{2}$ :
$X_{2} = X_{1} + X_{0} = 5 + 3 = 8$

STEP 8

Find $X_{3}$ :
$X_{3} = X_{2} + X_{1} = 8 + 5 = 13$

STEP 9

Find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 13 + 8 = 21$

STEP 10

And find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 21 + 13 = 34$

STEP 11

Now we have the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ as $\{3, 5, 8, 13, 21, 34, \ldots\}$ .

STEP 12

Notice the pattern that each sequence is simply the original sequence shifted one index forward. This means that the constants for the sequence are the Fibonacci numbers themselves.

STEP 13

Now, let's find the correct solution after indexing 5 steps ahead. This means we need to find the values of $X_{5}$ and $X_{6}$ in the original sequence to use as our new $X_{0}$ and $X_{1}$ .

STEP 14

From the original sequence, we already know that $X_{5} = 8$ and $X_{6} = 13$ .

STEP 15

Therefore, the new sequence starting with $X_{0} = 8$ and $X_{1} = 13$ after re-indexing 5 steps ahead would be $\{8, 13, 21, 34, 55, \ldots\}$ .

STEP 16

The correct solution to the difference equation after indexing 5 steps ahead is the sequence that starts with $X_{0} = 8$ and $X_{1} = 13$ .

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Redefine the Fibonacci sequence starting with $X_0=2, X_1=3$ or $X_0=3, X_1=5$ . Find the correct solution when the sequence is indexed 5 steps ahead.

STEP 1

STEP 2

First, let's find the solution for the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ .
We know that $X_{2} = X_{1} + X_{0}$ , so:
$X_{2} = 3 + 2 = 5$

STEP 3

Now, find $X_{3}$ using the difference equation:
$X_{3} = X_{2} + X_{1} = 5 + 3 = 8$

STEP 4

Next, find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 8 + 5 = 13$

STEP 5

We continue this process to find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 13 + 8 = 21$

STEP 6

Now we have the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ as $\{2, 3, 5, 8, 13, 21, \ldots\}$ .

STEP 7

Next, let's find the solution for the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ .
Again, we use the difference equation to find $X_{2}$ :
$X_{2} = X_{1} + X_{0} = 5 + 3 = 8$

STEP 8

Find $X_{3}$ :
$X_{3} = X_{2} + X_{1} = 8 + 5 = 13$

STEP 9

Find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 13 + 8 = 21$

STEP 10

And find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 21 + 13 = 34$

STEP 11

Now we have the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ as $\{3, 5, 8, 13, 21, 34, \ldots\}$ .

STEP 12

Notice the pattern that each sequence is simply the original sequence shifted one index forward. This means that the constants for the sequence are the Fibonacci numbers themselves.

STEP 13

Now, let's find the correct solution after indexing 5 steps ahead. This means we need to find the values of $X_{5}$ and $X_{6}$ in the original sequence to use as our new $X_{0}$ and $X_{1}$ .

STEP 14

From the original sequence, we already know that $X_{5} = 8$ and $X_{6} = 13$ .

STEP 15

Therefore, the new sequence starting with $X_{0} = 8$ and $X_{1} = 13$ after re-indexing 5 steps ahead would be $\{8, 13, 21, 34, 55, \ldots\}$ .

STEP 16

The correct solution to the difference equation after indexing 5 steps ahead is the sequence that starts with $X_{0} = 8$ and $X_{1} = 13$ .

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Redefine the Fibonacci sequence starting with X0=2,X1=3X_0=2, X_1=3X0​=2,X1​=3 or X0=3,X1=5X_0=3, X_1=5X0​=3,X1​=5. Find the correct solution when the sequence is indexed 5 steps ahead.

STEP 1

STEP 2

First, let's find the solution for the sequence starting with X0=2X_{0} = 2X0​=2 and X1=3X_{1} = 3X1​=3.We know that X2=X1+X0X_{2} = X_{1} + X_{0}X2​=X1​+X0​, so:X2=3+2=5X_{2} = 3 + 2 = 5X2​=3+2=5

STEP 3

Now, find X3X_{3}X3​ using the difference equation:X3=X2+X1=5+3=8X_{3} = X_{2} + X_{1} = 5 + 3 = 8X3​=X2​+X1​=5+3=8

STEP 4

Next, find X4X_{4}X4​:X4=X3+X2=8+5=13X_{4} = X_{3} + X_{2} = 8 + 5 = 13X4​=X3​+X2​=8+5=13

STEP 5

We continue this process to find X5X_{5}X5​:X5=X4+X3=13+8=21X_{5} = X_{4} + X_{3} = 13 + 8 = 21X5​=X4​+X3​=13+8=21

STEP 6

Now we have the sequence starting with X0=2X_{0} = 2X0​=2 and X1=3X_{1} = 3X1​=3 as {2,3,5,8,13,21,…}\{2, 3, 5, 8, 13, 21, \ldots\}{2,3,5,8,13,21,…}.

STEP 7

Next, let's find the solution for the sequence starting with X0=3X_{0} = 3X0​=3 and X1=5X_{1} = 5X1​=5.Again, we use the difference equation to find X2X_{2}X2​:X2=X1+X0=5+3=8X_{2} = X_{1} + X_{0} = 5 + 3 = 8X2​=X1​+X0​=5+3=8

STEP 8

Find X3X_{3}X3​:X3=X2+X1=8+5=13X_{3} = X_{2} + X_{1} = 8 + 5 = 13X3​=X2​+X1​=8+5=13

STEP 9

Find X4X_{4}X4​:X4=X3+X2=13+8=21X_{4} = X_{3} + X_{2} = 13 + 8 = 21X4​=X3​+X2​=13+8=21

STEP 10

And find X5X_{5}X5​:X5=X4+X3=21+13=34X_{5} = X_{4} + X_{3} = 21 + 13 = 34X5​=X4​+X3​=21+13=34

STEP 11

Now we have the sequence starting with X0=3X_{0} = 3X0​=3 and X1=5X_{1} = 5X1​=5 as {3,5,8,13,21,34,…}\{3, 5, 8, 13, 21, 34, \ldots\}{3,5,8,13,21,34,…}.

STEP 12

Notice the pattern that each sequence is simply the original sequence shifted one index forward. This means that the constants for the sequence are the Fibonacci numbers themselves.

STEP 13

Now, let's find the correct solution after indexing 5 steps ahead. This means we need to find the values of X5X_{5}X5​ and X6X_{6}X6​ in the original sequence to use as our new X0X_{0}X0​ and X1X_{1}X1​.

STEP 14

From the original sequence, we already know that X5=8X_{5} = 8X5​=8 and X6=13X_{6} = 13X6​=13.

STEP 15

Therefore, the new sequence starting with X0=8X_{0} = 8X0​=8 and X1=13X_{1} = 13X1​=13 after re-indexing 5 steps ahead would be {8,13,21,34,55,…}\{8, 13, 21, 34, 55, \ldots\}{8,13,21,34,55,…}.

STEP 16

The correct solution to the difference equation after indexing 5 steps ahead is the sequence that starts with X0=8X_{0} = 8X0​=8 and X1=13X_{1} = 13X1​=13.

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Redefine the Fibonacci sequence starting with $X_0=2, X_1=3$ or $X_0=3, X_1=5$ . Find the correct solution when the sequence is indexed 5 steps ahead.

First, let's find the solution for the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ .
We know that $X_{2} = X_{1} + X_{0}$ , so:
$X_{2} = 3 + 2 = 5$

Now, find $X_{3}$ using the difference equation:
$X_{3} = X_{2} + X_{1} = 5 + 3 = 8$

Next, find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 8 + 5 = 13$

We continue this process to find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 13 + 8 = 21$

Now we have the sequence starting with $X_{0} = 2$ and $X_{1} = 3$ as $\{2, 3, 5, 8, 13, 21, \ldots\}$ .

Next, let's find the solution for the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ .
Again, we use the difference equation to find $X_{2}$ :
$X_{2} = X_{1} + X_{0} = 5 + 3 = 8$

Find $X_{3}$ :
$X_{3} = X_{2} + X_{1} = 8 + 5 = 13$

Find $X_{4}$ :
$X_{4} = X_{3} + X_{2} = 13 + 8 = 21$

And find $X_{5}$ :
$X_{5} = X_{4} + X_{3} = 21 + 13 = 34$

Now we have the sequence starting with $X_{0} = 3$ and $X_{1} = 5$ as $\{3, 5, 8, 13, 21, 34, \ldots\}$ .

Now, let's find the correct solution after indexing 5 steps ahead. This means we need to find the values of $X_{5}$ and $X_{6}$ in the original sequence to use as our new $X_{0}$ and $X_{1}$ .

From the original sequence, we already know that $X_{5} = 8$ and $X_{6} = 13$ .

Therefore, the new sequence starting with $X_{0} = 8$ and $X_{1} = 13$ after re-indexing 5 steps ahead would be $\{8, 13, 21, 34, 55, \ldots\}$ .

The correct solution to the difference equation after indexing 5 steps ahead is the sequence that starts with $X_{0} = 8$ and $X_{1} = 13$ .