Răspuns
(a, b, c)∈{(3, 7, 11), (21, 7, -7)}
Explicație pas cu pas:
notam ratia =r si b=x
=> a=x-r, b=x si c=x+r
a+b+c=x-r+x+x+r=21
3x=21 => x=7
deci, a=7-r; b=7 si c=7+r
a+3=7-r+3=10-r
b+5=7+5=12
c+13=7+r+13=20+r
a+3, b+5, c+13 in progresie geometrica
=> (b+5)²=(a+3)(c+13)
144=(10-r)(20+r)
144=200-10r-r²
r²+10r-56=0
Δ=100+224=324
r1=(-10+18)/2=4
r2=(-10-18)/2=-14
r=4 => a=7-4=3; b=7; c=7+4=11
a=3; b=7; c=11 progresie aritmetica
a+3=6, b+5=12, c+13=24, sunt in progresie geometrica (144=6•24)
r= -14 => a=7+14=21; b=7; c=-7
=> a=21; b=7; c=-7 sunt in progresie aritmetica
a+3=24; b+5=12, c+13=-7+13=6; sunt in progresie geometrica (144=24•6)
(a, b, c)∈{(3, 7, 11), (21, 7, -7)}
