Maple problem - matrices and sequences

[literacola]

Maple problem -- matrices and sequences

Hello there Maple geniuses!

So I've been banging my head against these two problems for days and I was wondering if any of you had any idea how to proceed.

The first involves computing the limit of a series. The problem reads as follows:

"For large n, the sequence defined by

x1 = 2.30

xn = 9 * sqrt(xn-1+ 6)

approaches a limit. Find that limit correct to 15 significant figures, and then select the correct answer below."


The answer is supposedly 86.611... but doesn't this approach infinity? I've attempted plotting the sequence out and computing many iterations of the sequence and as expected, the only thing the sequence does is blow up larger and larger. How can this possibly approach 86? What am I missing here?


My second problem deals with matrices and the problem is stated as follows:

For the matrix T given below, find T to the power of 1000 and then multiply it on the right by the column vector <1,0,0,0,0,0,0,0>.

T := < < 9/100, 19/100, 0, 1/50, 1/100, 0, 27/50, 3/20 > |

< 0, 0, 1/25, 13/100, 1/4, 1/10, 1/100, 47/100 > |

< 27/100, 1/10, 13/25, 0, 1/50, 1/100, 2/25, 0 > |

< 0, 9/50, 21/50, 0, 13/100, 1/10, 1/100, 4/25 > |

< 13/20, 0, 0, 0, 1/100, 3/100, 31/100, 0 > |

< 67/100, 23/100, 0, 1/50, 3/100, 0, 1/20, 0 > |

< 0, 0, 0, 33/100, 23/50, 3/100, 0, 9/50 > |

< 0, 7/10, 0, 0, 1/100, 1/25, 1/4, 0 > >;


So I assigned T to the matrix as shown above, and ran "PT := T^1000" to raise it to the 1000th power and assign it to the variable TP. Then to multiply against the column vector matrix V, I ran "TP.V". This gives me this huge list of numbers, and nothing that even resembles a matrix. The answer is supposedly the column vector <0.157,0.166,0.0872,0.141,0.0413,0.177,0.147> but I don't get anything near that.


Can anybody pleassssee help?

 

[mighty2000]

Hi there! I can definitely understand your frustration with these problems. Let's take a closer look at each one and see if we can figure out a solution.

For the first problem, it may seem counterintuitive that the given sequence approaches a finite limit instead of approaching infinity. However, as n approaches infinity, the term within the square root will become smaller and smaller, causing the entire sequence to approach a finite limit. To solve for this limit, you can use Maple's "limit" function to find the limit of the expression as n approaches infinity. Once you have the limit, you can use the "evalf" function to round it to 15 significant figures.

For the second problem, it looks like you are on the right track with finding T to the power of 1000. However, instead of using the "TP.V" command, try using the "MatrixMultiply" function to multiply the resulting matrix by the given column vector. This should give you the correct answer.

I hope this helps! Don't hesitate to reach out if you have any further questions. Good luck!