āĻ¯āĻĻāĻŋ \(a:b=b:c\) āĻšā§, āĻ¤āĻŦā§ āĻĒā§āĻ°āĻŽāĻžāĻ¨ āĻāĻ°āĻŋ āĻ¯ā§, \(a^2b^2c^2\left(\cfrac{1}{a^3} +\cfrac{1}{b^3} +\cfrac{1}{c^3}\right)=a^3+b^3+c^3\)
Loading content...
\(a, b, c\) āĻā§āĻ°āĻŽāĻŋāĻ āĻ¸āĻŽāĻžāĻ¨ā§āĻĒāĻžāĻ¤ā§ āĻšāĻ˛ā§ \(a:b=b:c\)
āĻ§āĻ°āĻŋ, \(\cfrac{a}{b}=\cfrac{b}{c}=k (k \ne 0)\)
āĻ¸ā§āĻ¤āĻ°āĻžāĻ, \(a=bk=ck^2\) āĻāĻŦāĻ \(b=ck\)
āĻŦāĻžāĻŽāĻĒāĻā§āĻˇ \(=a^2 b^2 c^2 \left(\cfrac{1}{a^3} +\cfrac{1}{b^3} +\cfrac{1}{c^3}\right)\)
\(=(ck^2 )^2.(ck)^2.c^2 \left(\cfrac{1}{(ck^2 )^3} +\cfrac{1}{(ck)^3} +\cfrac{1}{c^3} \right)\)
\(=c^2 k^4. c^2 k^2.c^2\left(\cfrac{1}{c^3 k^6}+\cfrac{1}{c^3 k^3}+\cfrac{1}{c^3}\right)\)
\(=c^6 k^6.\cfrac{1}{c^3}\left(\cfrac{1}{k^6} +\cfrac{1}{k^3} +1\right)\)
\(=c^3 k^6 \left(\cfrac{1+k^3+k^6}{k^6} \right)\)
\(=c^3 (1+k^3+k^6 )=c^3 (k^6+k^3+1)\)
āĻĄāĻžāĻ¨āĻĒāĻā§āĻˇ \(=a^3+b^3+c^3\)
\(=(ck^2 )^3+(ck)^3+c^3\)
\(=c^3 k^6+c^3 k^3+c^3\)
\(=c^3 (k^6+k^3+1) \)
â´āĻŦāĻžāĻŽāĻĒāĻā§āĻˇ=āĻĄāĻžāĻ¨āĻĒāĻā§āĻˇ (āĻĒā§āĻ°āĻŽāĻžāĻ¨āĻŋāĻ¤)