Three members GP

The sum of three numbers in GP (geometric progression) is 21, and the sum of their squares is 189. Find the numbers.

Correct answer:

a =  3
b =  6
c =  12
a2 =  12
b2 =  6
c2 =  3

Step-by-step explanation:

a+b+c = 21 a2+b2+c2 = 189  b = qa c = q2a  a+qa+q2a = 21 a2 + q2a2+q4a2 = 189  a(1+q+q2) = 21 a2(1 + q2+q4) = 189  212(1 + q2+q4) = 189 (1+q+q2)2 252 q4  378 q3  126 q2  378 q + 252 = 0  q4 + q2 + 1 = (q2  q + 1)   (q2 + q + 1)  212(q2  q + 1)   (q2 + q + 1) = 189 (1+q+q2)2 212(q2q+1)=189(1+q+q2)  212(q2q+1)=189(1+q+q2) 252q2630q+252=0 252=22327 630=23257 GCD(252,630,252)=2327=126  2q25q+2=0  a=2;b=5;c=2 D=b24ac=52422=9 D>0  q1,2=2ab±D=45±9 q1,2=45±3 q1,2=1.25±0.75 q1=2 q2=0.5  a=21/(1+q1+q12)=21/(1+2+22)=3

Our quadratic equation calculator calculates it.

b=q1 a=2 3=6
c=q1 b=2 6=12
a2=21/(1+q2+q22)=21/(1+0.5+0.52)=12
b2=q2 a2=0.5 12=6
c2=q2 b2=0.5 6=3



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