Question

Numerele a, b si c sunt in progresie aritmetica si au suma 21. Stiind ca numerele a+3, b+5 si c+13 sunt in progresie geometrica, sa se determine a, b si c.

Answer 1

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)}