Question

Q. 5. If a, b, c and d are in continued proportion,

prove that:
a3+b3+c3/(b3 +c3+ d3) = a/d​

Answer 1

Step-by-step explanation:

Given:-

a, b, c and d are in continued proportion.

To find:-

prove that: a3+b3+c3/(b3 +c3+ d3) = a/d

Solution:-

a,b,c and d are in the continued proportion

=>a/b=b/c=c/d

Let a/b=b/c=c/d=k

a/b=k=>a=bk----------------(1)

b/c=k=>b=ck----------------(2)

c/d=k=>c=dk----------------(3)

from (2)&(3)

=>b=(dk)k=dk²

b=dk²-------------------------(4)

from(1)&(4)

a=(dk²)k=dk³

a=dk³-------------------------(5)

LHS:-

(a³+b³+c³)/(b³+c³+d³)

=>[(dk³)³+(dk²)³+(dk)³]/[(dk²)³+(dk)³+(d)³]

=>(d³k⁹+d³k⁶+d³k³)/(d³k⁶+d³k³+d³)

=>d³(k⁹+k⁶+k³)/d³(k⁶+k³+1)

=>(k⁹+k⁶+k³)/(k⁶+k³+1)

=>k³(k⁶+k³+1)/(k⁶+k³+1)

=>k³

(a³+b³+c³)/(b³+c³+d³)=k³-----(6)

RHS:-

a/d

=>dk³/d

=>k³

a/d=k³-----------------------------(7)

From(6)&(7)

LHS=RHS

(a³+b³+c³)/(b³+c³+d³)=a/d

Hence, Proved