first two terms is 12 and the third term is 16 . (a) Determine the ratio \( \frac{a r^{2}}{a+a r} \) (b) If the first term is larger than the second term, find the value of r .

\( \left(I^{\text {st }}\right. \) Session 2022) If the ratio between the fourth term from the beginning to the fourth term from the end in the expansion of \( \left(\frac{a}{x}+\frac{x}{a}\right)^{2 n} \) equals \( a^{4}: x^{4} \) , then \( n=\cdots \cdots \) \( \begin{array}{llll}\text { (a) } 2 & \text { (b) } 4 & \left.\text { (where } x \in \mathbb{R}^{*}, a \in \mathbb{R}^{*}\right)\end{array} \)

Solve for \( a \) in the proportion. \[ \begin{array}{l}\frac{19}{a}=\frac{9.5}{6} \\ a=\square\end{array} \]

In the expansion \( \left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n} \) if the ratio between the seventh term from the beginning to the seventh term from the end \( 1: 6 \), then \( n \) equals \( \begin{array}{llll}\text { (a) } 7 & \text { (b) } 8 & \text { (c) } 9 & \text { (d) } 10\end{array} \)

Soient xety deux nombres reels tels que \( : x \leq \frac{1}{2}, y \geq \frac{3}{2} \) et \( x-2 y=5 \) On pose \( A=\sqrt{x^{2}+(x-1)^{2}+2 x(x-1)} \) et \( B=\sqrt{4 y^{2}+(2 y-3)^{2}+4 y(2 y-3)} \) 1) Montrer que : \( A=\sqrt{(2 x-1)^{2}} \) et \( B=\sqrt{\left(4 y-3 t^{2}\right.} \) 4) Simplifier Ies nombres \( A \) et \( B \) 8) Determiner la valeur numérique du nombre \( A+3 \) X

Solve the equation: \( \sqrt{x + 4} = 6 \)