Question Number 70040 by Shamim last updated on 30/Sep/19

If, a^2 b^2 c^2 ((1/a^3 )+(1/b^3 )+(1/c^3 ))=a^3 +b^3 +c^3  than  prove that a,b,c Successive Proportional.

Commented byPrithwish sen last updated on 01/Oct/19

(((b^2 c^2 )/a)−a^3 )+(((a^2 c^2 )/b)−b^3 )+(((a^2 b^2 )/c) − c^3 )=0  ((b^2 c^2 −a^4 )/a) +((a^2 b^2 −c^4 )/c) +(((a^2 c^2 −b^4 )/c))=0  ((b^2 c^3 −a^4 c + a^3 b^2 −ac^4 )/(ac)) +((a^2 c^2  − b^4 )/c) = 0  (((b^2 −ac)(a^3 + c^3 ))/(ac)) − (((b^2 −ac)(b^2 +ac))/c) = 0  (b^2 −ac)(((a^3 +c^3 )/(ac)) − ((b^2 +ac)/c) )=0   considering only                                  b^2 −ac = 0  ⇒ a,b,c are in proportion.  proved

Commented byShamim last updated on 01/Oct/19


Commented byMJS last updated on 01/Oct/19

of course this is right    I saw that the given equation can be  transformed to  a^4 bc+ab^4 c+abc^4 −a^3 b^3 −a^3 c^3 −b^3 c^3 =0 ⇔  ⇔ (a^2 −bc)(b^2 −ac)(c^2 −ab)=0  but I thought it′s not easy to factorize    on the other hand by taking the result and  simply putting b=pa, c=p^2 a or just c=(b^2 /a) we  easily find the given equation is true    but I thought showing that with b=pa and  c=qa ⇒ q=p^2  is more convincing

Answered by MJS last updated on 30/Sep/19

for a, b, c ∈R we can find p, q ∈R ⇒ b=pa∧c=qa    a^2 b^2 c^2 ((1/a^3 )+(1/b^3 )+(1/c^3 ))=a^3 +b^3 +c^3   a^3 b^3 +a^3 c^3 +b^3 c^3 =a^4 bc+ab^4 c+abc^4   a^4 bc−a^3 (b^3 −c^3 )+a(b^4 c+bc^4 )−b^3 c^3 =0  b=pa∧c=qa, let a≤b≤c ⇒ 1≤p≤q  a^6 (−p^3 q^3 +p^4 q+pq^4 −p^3 −q^3 +pq)=0  a≠0  −p^3 q^3 +p^4 q+pq^4 −p^3 −q^3 +pq=0  (p^2 −q)(p−q^2 )(pq−1)=0  ⇒ q=p^2   [∨q=±(√p)∨q=(1/p)]  ⇒ ⟨a, b, c⟩=⟨a, pa, p^2 a⟩

Commented byShamim last updated on 30/Sep/19

It′s very hard. others rule???

Commented byMJS last updated on 30/Sep/19

it′s not that hard. write  −p^3 q^3 +p^4 q+pq^4 −p^3 −q^3 +pq=0  as  pq^4 −(p^3 +1)q^3 +p(p^3 +1)q−p^3 =0  divide by p  q^4 −((p^3 +1)/p)q^3 +(p^3 +1)q−p^2 =0  we know if there′s an easy solution q_1  ⇒  q_1 ∣p^2  ⇒ we have to try ±1, ±p, ±p^2   ⇒ q_1 =p^2  and this is enough because we  set a≤b≤c ⇒ 1≤p≤q ⇒ q≠±(√p)∧q≠(1/p)    we then have  (q−p^2 )(q^3 −(1/p)q^2 −pq+1)=0  transform to  (q−p^2 )(q^2 (q−(1/p))−p(q−(1/p)))=0  ⇒ q_2 =(1/p)  (q−p^2 )(q−(1/p))(q^2 −p)=0  ⇒ q_(3, 4) =±(√p)