Question

Dau coroana !!!! Va rog, este urgent!!! sa se determine numerele reale in progresie geometrică a, b, c, daca suma lor este 26, iar numerele a+1, b+6, c+3 sunt in progresie aritmetica

Answer 1

a+1; b+6; c+3

$b+6= \frac{a+1+c+3}{2} \Rightarrow b = \frac{a+c+4}{2}-6 \Rightarrow b = \frac{a+c-8}{2} \\ \\ a+b+c =26 \Rightarrow a+ \frac{a+c-8}{2}+c = 26\Rightarrow 2a+a+c-8+c = 52\Rightarrow \\ \Rightarrow 3a+3c = 60 \Rightarrow3(a+c) = 60 \Rightarrow a+c = 20 \\ \\ a+c = 20 \\ a+c = 26-b \\ \Rightarrow 26-b = 20 \Rightarrow b = 6$

a; b; c (in prog, geometrica)

$b = \sqrt{a*c} \Rightarrow \sqrt{a*c} =6 \Rightarrow a*c = 36 \Rightarrow a = \frac{36}{c} \\ a+c=20 \\ \frac{36}{c} +c = 20 \Rightarrow 36+ c^{2} = 20c \Rightarrow c^{2} -20c +36 = 0 \\ \Delta = 400-144 =256 = 16^{2} \\ \\ c_{1,2}= \frac{20\pm16}{2} \Rightarrow \left \{ {{ c_{1} =3} \atop { c_{2} =18}} \right. \\ \\ I) c = 3 \Rightarrow a+6+3 = 26 \Rightarrow a = 26-9 \Rightarrow a = 17 \Rightarrow 17;6;3 (F) \\ II)c=18 \Rightarrow a+6+18 = 26\Rightarrow a = 26-24 \Rightarrow a = 2$

=> 2;6;18 (progresia aceasta convine)

=> Solutiile sunt: a = 2; b = 6; c = 18