যদি \(\cfrac{a}{b}=\cfrac{c}{d}=\cfrac{e}{f}\) হয়, তবে প্রমাণ করো প্রত্যেকটি অনুপাত = \(\left(\cfrac{la^n+mc^n+pe^n}{lb^n+md^n+pf^n}\right)^{\cfrac{1}{n}}\)
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ধরি, \(\cfrac{a}{b}=\cfrac{c}{d}=\cfrac{e}{f}=k\)
\(\therefore a=bk, c=dk\) এবং \(e=fk\)
\(\therefore \left(\cfrac{la^n+mc^n+pe^n}{lb^n+md^n+pf^n}\right)^{\cfrac{1}{n}}\)
\(=\left(\cfrac{lb^nk^n+md^nk^n+pf^nk^n}{lb^n+md^n+pf^n}\right)^{\cfrac{1}{n}}\)
\(=\left(\cfrac{k^n\cancel{[lb^n+md^n+pf^n]}}{\cancel{[lb^n+md^n+pf^n]}}\right)^{\cfrac{1}{n}}\)
\(=(k^n)^{\cfrac{1}{n}}\)
\(=k\)
\(\therefore \cfrac{a}{b}=\cfrac{c}{d}=\cfrac{e}{f}=\left(\cfrac{la^n+mc^n+pe^n}{lb^n+md^n+pf^n}\right)^{\cfrac{1}{n}}\) [উভয়ের মান-ই \(k\)] (প্রমানিত)