341CS kills it at sight
a_{1}^{2}+ a_{2}^{2}+a_{3}^{2}= 25
b_{1}^{2}+b_{2}^{2}+b_{3}^{2}=100
b_{1}a_{1}+b_{2}a_{2}+b_{3}a_{3}= 50
Find \; \; \frac{a_{1}^{n}+a_{2}^{n}+a_{3}^{n}}{b_{1}^{n}+ b_{2}^{n}+b_{3}^{n}}
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10 Answers
1Let \ \vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \\ \ \ and\ \vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}
u'll notice that a || b
Hence\ \; \; \frac{a_{1}^{n}+a_{2}^{n}+a_{3}^{n}}{b_{1}^{n}+ b_{2}^{n}+b_{3}^{n}}=2^{-n}
1@ sir : CS -- Counter Strike ??
didnt get ?
anyways i think my soln is okay....
but pls tell your method
1Oh if it is Cauchy schwarz that you meant then Im ok even without seeing that soln...
1@Nishant Bhaiyya : yes, I noticed just after i wrote that this could be a nice proof for CS. atleast in exact such form for three variables.
However i think it could be extended to any number of variables by induction.
The thought of extending such a nice one by induction was definitely not as satisfying ..so i had a vague idea that something like an n-dimensional vector should exist and sure enough (as i came to know some time later) it does.