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Posts Tagged ‘ analysis ’

Limit Preserving Functions in CPOs

Very short proof that limit preserving functions (continuous functions) on complete partial orders are necessarily monotone.

14 August
Posted in Article, Math Problems

Even Pascals

Let a=4k-1, where k is an integer. Prove that for any positive integer n the number

     $$ s_n = 1 - {n \choose 2}a + {n \choose 4}a^2 - {n \choose 6}a^3 + \cdots $$

is divisible by 2^{n-1}.

14 August
Posted in Article, Math Problems

Sequences in Sequences

Define the sequence (a_n)_n recursively by a_1 = 1 and

     $$ a_{n+1} = \frac{1 + 4a_n + \sqrt{1+24a_n}}{16} ,\hspace{4mm} \mbox{for $n \ge 1$.} $$

Find an explicit formula for a_n in terms of n.

14 August
Posted in Article, Math Problems

More Linear Recurrences

Let (x_n )_n = 0 be defined by the recurrence relation x_{n + 1} = ax_n + bx_{n - 1}, with x_0 = 0. Show that the expression x^2_n - x_{n - 1} x_{n + 1} depends only on b and x_1, but not on a.

13 August
Posted in Article, Math Problems

Almost Linear

Find the general term of the sequence given by x_0 = 3, x_1 = 4, and

    $$(n + 1 )(n + 2 )x_n = 4 (n + 1 )(n + 3 )x_{n - 1} - 4 (n + 2 )(n + 3 )x_{n - 2}$$

13 August
Posted in Article, Math Problems

Polynomial Divisors

Let p(x) = x^2 -3x + 2. Show that for any positive integers  n \ge 2 there exists unique numbers a_n, b_n such that the polynomial  x^n - a_n x - b_n is divisible by p(x)

13 August
Posted in Article, Math Problems

Analytical Fibonacci

We derive Binet’s equation for the nth Fibonacci number as

     $$ F_n = \frac{1}{\sqrt 5}\left( \left(\frac{1+\sqrt 5}{2}\right)^n -  \left(\frac{1-\sqrt 5}{2}\right)^n  \right) $$

13 August
Posted in Article, Math Problems

Linear Recursive Sequence

We derive a general technique for solving full rank linear recursive sequences. Formally, a kth linear recursive sequence is defined as

     $$ x_n = a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3} + \stackrel{k}{\ldots} + a_k x_{n-k} $$

8 August
Posted in Article, Math Problems

The Fib-Fib-Fib Sequence

Consider the sequences (a_n)_n, (b_n)_n, defined recursively by

     $$ \begin{tabular}{l l l l} $a_0 = 0$, & $a_1 = 2$, & $a_{n+1} = 4a_n + a_{n-1}$ & $n \ge 0$ \\ $b_0 = 0$, & $b_1 = 1$, & $b_{n+1} = a_n - b_n + b_{n-1}$ & $n \ge 0$ \end{tabular} $$

Show that (a_n)^3 = b_{3n} \fa n

7 August
Posted in Article, Math Problems

Square Sequence

The sequence f_1, f_2, \cdots satisfies

    $$ f_{m+n} + f_{m - n} = \frac{f_{2m} + f_{2n}}{2}$$

for all nonnegative integers n,m and m \ge n. If f_1 = 1, determine f_n