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Pe: 2009-10-31T13:46:17+02:00 In: In:

Progresii

1) Fie progresia aritmetica (an)n ≥1 cu ratia r. Stiind ca:
a) a3 = 8 + a4=5. Calculati a1 si ratia;
b) a2 = a6+a4=-7 si a8 – a7 =2 a 4. Calculati a1 si r

2) Fie ca progresia geometrica (bn)n≥1 cu ratia q. Stiind ca:
a) b1 – b2 =8 si b2 + b3 = 12. Aflati b1 si q.

3) Sa se arate ca numerele a patrat ,b patrat ,c patrat sunt in progresie aritmetica atunci si numerele a patrat – bc, b patrat – ca, c patrat – ab sunt in progresie aritmetica.

Similare

3 raspunsuri

  1. maestru (V)

    a3 = 8 + a4=5. Calculati a1 si ratia


    Rezolvare

        \[ \begin{array}{l} {\rm{a}}_{\rm{3}} {\rm{ }} = {\rm{ 8 }} + {\rm{ a}}_{\rm{4}} = {\rm{5}} \Rightarrow *\left\{ \begin{array}{l} {\rm{a}}_{\rm{3}} {\rm{ }} = {\rm{ 8 }} + {\rm{ a}}_{\rm{4}} \\ {\rm{8 }} + {\rm{ a}}_{\rm{4}} = {\rm{5}} \\ \end{array} \right. \\ {\rm{a}}_{\rm{3}} = {\rm{a}}_1 + 2r \\ {\rm{a}}_{\rm{4}} = {\rm{a}}_1 + 3r \\ * \Leftrightarrow \left\{ \begin{array}{l} {\rm{a}}_1 + 2r = 8 + {\rm{a}}_1 + 3r \\ {\rm{8 + a}}_1 + 3r = 5 \\ \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} r = - 8 \\ {\rm{8 + a}}_1 + 3r = 5 \\ \end{array} \right. \Rightarrow {\rm{a}}_1 = - 3 + 24 = 21 \\ \end{array} \]

    b) se rezolva asemanator.

    • 0
  2. maestru (V)

    Fie ca progresia geometrica (bn)n≥1 cu ratia q. Stiind ca:
    b1 – b2 =8 si b2 + b3 = 12. Aflati b1 si q.


    Rezolvare:

        \[ \begin{array}{l} *\left\{ \begin{array}{l} {\rm{b}}_{\rm{1}} - {\rm{b}}_{\rm{2}} = {\rm{8}} \\ {\rm{b}}_{\rm{2}} + {\rm{b}}_{\rm{3}} = {\rm{12}} \\ \end{array} \right. \\ {\rm{b}}_{\rm{2}} = b_1 \cdot q \\ {\rm{b}}_{\rm{3}} = b_1 \cdot q^2 \\ * \Leftrightarrow \left\{ \begin{array}{l} {\rm{b}}_{\rm{1}} - b_1 \cdot q = 8 \Rightarrow b_1 = \frac{8}{{1 - q}} \\ b_1 \cdot q + b_1 \cdot q^2 = 12 \\ \end{array} \right. \\ \frac{{8q}}{{1 - q}} + \frac{{8q^2 }}{{1 - q}} = 12 \Leftrightarrow 12 - 12q = 8q + 8q^2 \Rightarrow \\ 8q^2 + 20q - 12 = 0 \Rightarrow \left\{ \begin{array}{l} q_1 = - 3 \\ q_2 = \frac{1}{2} \\ \end{array} \right. \\ \end{array} \]

    Avand q se determina imediat b1.

    • -1
  4. user (0)

    Multumesc din suflet!

    Ma poate auta cineva si la exercitiul 3?

    • 0
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