Question

suma a trei numere in progresie geometrica crescatoare este 26. daca la aceste numere se adauga 1,6 respectiv 3,se formeaza o progresie aritmetica.aflati numerele



va rog sa ma ajutati,e urgent

Answer 1

Răspuns:

numerele sunt

2;6;18 cu q=3

Explicație pas cu pas:

fie x, xq si , respectiv xq² cele trei numere

atunci

x(1+q+q²)=26

x+1+xq²+3=2 (xq+6)

x+xq+xq²=26

x-2xq+xq²=8

fie x+xq²=u

xq=v

u+v=26

u-2v=8

v=26-u

u-2(26-u)=8

3u=52+8=60

u=20

v=26-20=6

x+xq²=20

xq=6

(1+q²)/q=20/6=10/3

3q²+3=10q

3q²-10q+3=0

q1,2=(10±√(100-36))/(2*3)= (10±8)/6n

q1=1/3 <1, nu convine

q2=3>1 , convine

xq=x*3=6...x=2

numerele sunt

2;6;18 cu q=3

care ar forma

3;12;21 unde r=9

Answer 2

$\it Fie\ a,\ aq,\ aq^2\ cele\ trei\ numere,\ ale\ progresiei\ geometrice .\\ \\ a + aq + aq^2 = 26\ \ \ \ \ (1)\\ \\ a+1,\ aq+6,\ aq^2+3\ formeaz\breve{a}\ o\ progresie\ aritmetic\breve{a}.\\ \\ Suma\ lor\ este:\\ \\ S_3=a+1+aq+6+aq^2+3=(a+aq+aq^2)+10\ \stackrel{(1)}{=}\ 26+10=36\ \ \ \ \ (2)$

$\it Dar,\ conform\ formulei\ de\ la\ progresii\ aritmetice, avem:\\ \\ S_3=\dfrac{(a+1+aq^2+3)\cdot3}{2}=\dfrac{(a+aq^2+4)\cdot3}{2}\ \ \ \ \ (3)\\ \\ (2),\ (3) \Rightarrow \dfrac{(a+aq^2+4)\cdot3}{2}=36|_{\cdot\dfrac{2}{3}}\Rightarrow a+aq^2+4=24|_{-4} \Rightarrow \\ \\ \Rightarrow a+aq^2=20\ \ \ \ \ (4)$

$\it (1),\ (4)\Rightarrow 20+aq=26|_{-20} \Rightarrow aq=6\Rightarrow a=\dfrac{6}{q}\ \ \ \ \ (5)$

$\it (1),\ (5)\Rightarrow \dfrac{6}{q}+6+6q=26|_{-6}\Rightarrow \dfrac{6}{q}+6q=20|_{\cdot q}\Rightarrow 6+6q^2=20q\Rightarrow\\ \\ \Rightarrow6q^2-20q+6=0|_{:2}\Rightarrow3q^2-10q+3=0\Rightarrow3q^2-9q-q+3=0\Rightarrow\\ \\ 3q(q-3)-(q-3)=0\Rightarrow (q-3)(3q-1)=0\Rightarrow q_1=\dfrac{1}{3},\ \ q_2=3$

$\it q=\dfrac{1}{3}\ \stackrel{(5)}{\Longrightarrow}\ a=\dfrac{6}{\dfrac{1}{3}}=6\cdot3=18\Rightarrow progresia\ geometric\breve{a}:\\ \\ 18,\ \ 6,\ \ 2,\ ...\ \ descresc\breve{a}toare\ (nu\ convine).\\ \\ q=3\ \stackrel{(5)}{\Longrightarrow}\ a=\dfrac{6}{3}=2\Rightarrow progresia\ geometric\breve{a}:\\ \\ 2,\ \ 6,\ \ 18,\ ...\ \ cresc\breve{a}toare\ (\ convine !\ ).$

Prin urmare, numerele cerute sunt:  2,  6,  18.