5. \( \frac{(n+1)!}{(n-4)!}=6 n^{2}+6 n \) olduğuna göre, \( n \) kaçtır? \( \begin{array}{llll}\text { A) } 7 & \text { B) } 6 & \text { C) } 5 & \text { D) } 4 \\ \text { 5) } 3\end{array} \)

14 Expressa el vector \( \overrightarrow{\mathrm{a}}(-1,-8) \) com a combinació lineal de \( \overrightarrow{\mathrm{b}}(3,-2) \) i \( \overrightarrow{\mathrm{c}}\left(4,-\frac{1}{2}\right) \).

1. Find an odd natural number \( x \) such that \( \operatorname{LCM}(x, 40)=1400 \). 2. There are between 50 and 60 number of eggs in a basket. When Loza counts by 3 's, there are 2 eggs left over. When she counts by 5 's, there are 4 left over. How many eggs are there in the basket? 3. The GCF of two numbers is 3 and their LCM is 180 . If one of the numbers is 45 , then find the second number. 4. Using Mathematical Induction, prove the following: a) \( 6^{n}-1 \) is divisible by 5 , for \( n \geq 0 \). b) \( 2^{n} \leq(n+1)!, \quad \) for \( n \geq 0 \) c) \( x^{n}+y^{n} \) is divisible by \( x+y \) for odd natural number \( n \geq 1 \). d) \( 2+4+6+\cdots+2 n=n(n+1) \) e) \( 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6} \) 47 f) \( 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\frac{n^{2}(n+1)^{2}}{4} \) g) \( \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots+\frac{1}{n(n+1)}=\frac{n}{n+1} \)

5.) \( \begin{array}{l}\text { If } f(x)=2 x+1 \text { sad } f(g(x))=6 x-3 \\ \text { then } g(0)= \\ \begin{array}{llll}\text { (A) }-2 & \text { (B) }-1 & \text { (C) } 0 & \text { (D) } 1\end{array}\end{array} \) (E) 3

2) For the linear function \( f \), if \( f(2)=2 \) and \( f(4)=5 \), then for what value of \( x \) is \( f(x)=0 \) ? \( \begin{array}{lllll}\text { (A) } \frac{2}{3} & \text { (B) } \frac{3}{4} & \text { (C) } 1 & \text { (D) } \frac{4}{3} & \text { (E) } \frac{3}{2}\end{array} \)

rationalizing a denominator \( \frac{x}{\sqrt[3]{x^{4}}+\sqrt[3]{x^{2}}+1} \)