Question

If a:b=x:y=m:n then prove that
byn{(a+b)/b+(x+y)/y+(m+n)/n}³=27(a+b)(x+y)(m+n)

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Answer 1

Answer:

a:b=x:y=m:n=k (let)

So, a=bk, x=yk, m=nk

(a+b)/b=(bk+b)/b=b(k+1)/b=k+1

(x+y)/y=(yk+y)/y=y(k+1)/y=k+1

(m+n)/n=(mk+n)/n=n(k+1)/n=k+1

adding the above three we get,

a+b)/b+(x+y)/y+(m+n)/n=3(k+1)

cubing we get,

{a+b)/b+(x+y)/y+(m+n)/n}=27(k+1)^3

LHS = byn*27(k+1)^3

a+b=bk+b=b(k+1)

x+y=yk+y=y(k+1)

m+n=nk+n=n(k+1)

Multiplying we get,

(a+b)(x+y)(m+n)=byn(k+1)^3

Hence,

RHS= 27byn(k+1)^3=LHS (Proved)