54. Let A = {..., -2, -1, 0, 1, ..., i}. Find b) A | A a) UA₁. =

Let M = 10 6 [:: -3 1 Find formulas for the entries of M", where n is a positive integer. Mn = ((-2)(7^n))+4^n 7^n-(4^n) (-2)(7^n-4^n) 7^n-(2(4^n))

. Let $n \in \mathrm{N}^{*}$ and the equation:$$C_{2 n+1}^{1} x^{n}-C_{2 n+1}^{3} x^{n-1}+C_{2 n+1}^{5} x^{n-2}-\ldots+(-1)^{k} C_{2 n+1}^{2 k+1} x^{n-k}+\ldots+(-1)^{n} C_{2 n+1}^{2 n+1}=0$$Show that:a) the equation admits as roots the numbers::$\mathrm{x}_{\mathrm{k}}=\operatorname{ctg}^{2} \frac{k \pi}{2 n+1}$, for every $k \in\{1.2, \ldots . n\}$  b) $\frac{\pi^{2} n(2 n-1)}{3(2 n+1)^{2}}<\sum_{k=1}^{n} \frac{1}{k^{2}}<\frac{2 \pi^{2} n(n+1)}{3(2 n+1)^{2}}$ c) $\frac{\pi^{4} n(2 n-1)\left(4 n^{2}+10 n-9\right)}{45(2 n+1)^{4}}<\sum_{k=1}^{n} \frac{1}{k^{4}}<\frac{8 \pi^{4} n(n+1)\left(n^{2}+n+3\right)}{45(2 n+1)^{4}}$  d) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$  e) $\lim _{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{k^{4}}=\frac{\pi^{4}}{4}$  Please check the attached picture for details. I need complete solution with explanation please. the all fifth subpoints are for the same problem and they are conected between them, but if…

Find the number of arrangements of the digits {0,1,2,3,4,5,6,7,8,9} such that the first digit is less than 7 AND the second digit is greater than or equal to 2.

Two distinct numbers are selected from the set (1,2,3,4,...,36,37) so that the sum of the remaining 35 numbers is the product of these two numbers. What is the difference of these two numbers?

Let *' and '#' be two binary operations such that a*b=a^2 b and a # b = 2a+b. Find (2*3) # (6*7). *

why/where/how does the 1 and the 0 about from at the end of the last few steps?

Suppose a license plate number in this state can only contain capital letters and digits and can only be any string of the form: Either-Letter-Letter-Digit-Digit-Digit-Digit where "Either" means it can be either Capital letter or digit. How many different license plate numbers are possible? A (26^3)*(10^4) B (26^2)*(10^5) C (26+10)(26^2)(10^4) D (26+10)^2* (26^2)*(10^3)

How many zeroes are at the end of 25! (= 25 factorial)?

Let a, b and c be positive numbers such that a ^ 2 + b ^ 2 = c ^ 2Show that a + b> c.