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iwannalearnmath
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Pe: 2015-09-01T15:24:56+03:00 In: In:

Progresii . Ajutor va rog :)

1. Sa se calculeze suma primilor 20 de termeni ai unei progresii aritmetice , daca a6+a9+a12+a15=30.
2. Determinati ratia unei progresii geometrice daca a1+a4=7/16 , iar a1-a2+a3=7/8
3. Numerele reale a,b,c,d sunt in progresie geometrica. Daca d-a=7 , c-b=2 , aflati ratia progresiei

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  1. user (0)

    Buna seara
    Scriem:
    1)S_{20}=\dfrac{(a_1+a_{20})20}{2}=10(2a_1+19r)
    In acelasi timp prin ipoteza:
    (a_1+5r)+(a_1+8r)+(a_1+11r)+(a_1+15r)=30;2a_1+19r=15
    S_{20}=10\cdot 15=150.
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    2) \begin{cases} a_1+a_4&=\dfrac{7}{16}\\ a_1-a_2+a_3&=\dfrac{7}{8}\end{cases}\\ \begin{cases} a_1+a_1q^3&=\dfrac{7}{16}\\ a_1-a_1q+a_1q^2&=\dfrac{7}{8}\end{cases}\\ \begin{cases} a_1(1+q)(1-q+q^2)&=\dfrac{7}{16}\\ a_1(1-q+q^2)&=\dfrac{7}{8}\end{cases}\\ 1+q=\dfrac{7}{16}:\dfrac{7}{8}=\dfrac{1}{2};q=-\dfrac{1}{2}
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    3)\begin{cases} aq^3-a&=7\\ aq^2-aq&=2\end{cases}\\ \begin{cases} a(q-1)(q^2+q+1)&=7\\ aq(q-1)&=2\end{cases}\\ \dfrac{q^2+q+1}{q}=\dfrac{7}{2}\\ 2q^2-5q+2=0\ cu\ q\in{\{2,\dfrac{1}{2}}\}

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