Mathematics Department 12. During the recycling drive, \( \frac{1}{5} \) of the material collected was bottles and \( \frac{1}{4} \) was paper. Cardboard boxes made up \( \frac{1}{10} \) of the material. How much of the total do these three items represent?

Encuentie lai ewaciones simétricas de la recta que contiene ios puntos \( P(3,4,-1) \) Q \( (-2,4,6) \)

Soit \( n \) un entier naturel. On rappelle que pour tout réel \( q \) différent de 1, \[ \sum_{k=0}^{n} q^{k}=1+q+q^{2}+q^{3}+\ldots+q^{n}=\frac{1-q^{n+1}}{1-q} \] A l'aide de cette égalité, déterminer la limite de la suite \( \left(u_{n}\right) \) dans chacun des cas suivants. 1. \( u_{n}=1+\frac{1}{2}+\frac{1}{4}+\ldots+\frac{1}{2^{n}}=\sum_{k=0}^{n} \frac{1}{2^{k}} \) 2. \( u_{n}=1+\frac{2}{3}+\frac{4}{9}+\ldots+\frac{2^{n}}{3^{n}}=\sum_{k=0}^{n} \frac{2^{k}}{3^{k}} \) 3. \( u_{n}=8+2+\frac{1}{2}+\frac{1}{8}+\ldots+\frac{8}{4^{n}}=\sum_{k=0}^{n} \frac{8}{4^{k}} \)

Simplify \( x^{2}(x+1) \)

b) \( 2 \sqrt{72}-\sqrt{32} \)

80 Rappresenta nel piano cartesiano i punti \( A(5 ; 3), B(12 ; 3) \) e \( C(6 ; 8) \). Calcola l'area del triangolo \( A B C \).