a_i ,b_i ,x_i be reals for i=1,2,3,...,n, such that Σ_(i=1) ^n [a_i x_i ]=0. Prove that (Σ_(i=1) ^n [x_i ^2 ])(Σ_(i=1) ^n [a_i ^2 ]Σ_(i=1) ^n [b_i ^2 ]−(Σ_(i=1) ^n [a_i b_i ])^2 )≥(Σ_(i=1) ^n [a_i ^2 ])(Σ_(i=1) ^n [b_i x


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Question Number 195569 by York12 last updated on 05/Aug/23

$a_{i}, b_{i}, x_{i} b e r e a l s f o r i = 1, 2, 3, . . ., n, s u c h t h a t$ $\sum_{i = 1}^{n} [a_{i} x_{i}] = 0 . P r o v e t h a t$ $(\underset{i = 1}{\sum^{n}} [x_{i}^{2}]) (\underset{i = 1}{\sum^{n}} [a_{i}^{2}] \underset{i = 1}{\sum^{n}} [b_{i}^{2}] - {(\underset{i = 1}{\sum^{n}} [a_{i} b_{i}])}^{2}) ⩾ (\underset{i = 1}{\sum^{n}} [a_{i}^{2}]) {(\underset{i = 1}{\sum^{n}} [b_{i} x_{i}])}^{2}$
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