1) Determinati valorile expresiei E ( n) = ( -1) la puterea n+2 · ( -19) + ( -1) la puterea 3n+1 · ( - 23) , n ∈ N 2) In Δ dreptunghic ABC cu m(∧A) = 90 grade si m(∧C) = 30 grade , se construiesc AD inaltime si AM mediana , D , M ∈ ( BC) . Aratati ca : a) suprafetele ADM si ADB sunt echivalente ; b) suprafata ADB este jumatate din suprafata AMC ; c) DC= 3DB.

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2014-08-29T14:45:39+03:00
1)a)Daca n par=>n+2 par si 3n+1 impar=>(-1)^{n+2}=1;(-1)^{3n+1}=-1\\
E(n)=-19+23=4
b)Daca n impar=>n+2  impar si 3n+1 par=>(-1)^{n+2}=-1;(-1)^{3n+1}=1\\
E(n)=19-23=-4
2)a)AM mediana in triunghi dreptunghic=>AM=BC/2=BM=MC=>ΔABM echilateral si cum AD inaltime=>Ad mediana=>BD=DM
A_{ADM}= \frac{AD \cdot DM}{2}= \frac{AD \cdot DB}{2}=A_{ADB}
b)MC=MB=2BD
A_{AMC}=\frac{AD \cdot MC}{2}\\
A_{ADB}=\frac{AD \cdot BD}{2}\\
A_{ADB}= \frac{A_{AMC}}{2}
c)DC=DM+MC=BD+2DB=3DB.