2.2 The following information is given about a quadratic number pattem: \( T_{1}=3, T_{2}-T_{1}=9 \) and \( T_{3}-T_{2}=21 \) \( \begin{array}{ll}2.2 .1 & \text { Show that } T_{5}=111 \\ 2.2 .2 & \text { Show that the general term of the quadratic pattern is } T_{n}=6 n^{2}-9 n+6 \\ 2.2 .3 & \text { Show that the pattern is increasing for all } n \in N .\end{array} \)

Câu 03: (a) Rût gơn biểu thực \[ \left(\frac{1-x}{x^{2}+x^{3}-x^{4}}-\frac{x^{2}+x+2}{x^{5}-x^{3}-2 x^{2}-x}\right):\left(\frac{1+x^{4}}{x^{3}+x^{4}+x^{5}}-\frac{1-x+x^{2}}{x}\right) \] (b) Phân tích phân thức \( \frac{1}{(x-1)(x-2)(x-3)(x-4)} \) thành tổng các phân thức dơn giản.

\( f(x)=5 x+6 ; g(x)=3 x-2 \) (h) Find \( \left(\frac{1}{g}\right)(1) \).

Simplify he following fully: \( \frac{3^{3+1}-3^{-1}}{23^{x}} \) \( \frac{3^{x} \cdot 3^{1}-3^{x} \cdot 3^{-1}}{2 \cdot 3^{x}} \frac{3^{x}\left(1-\frac{1}{3}\right)}{23^{x}} \)

How much pure acid should be mixed with 2 gallons of a \( 50 \% \) acid solution to get an \( 80 \% \) acid solution?

QUESTION 12 If \( 2^{x} 5^{y}=16 \times 125 \), where \( x \) and \( y \) are positive integers, determine \( x-y \) :