a_h is Cauchy Sequence. Sequence {a_h }_(h=1) ^n Satisfy Σ_(h=1) ^n a_h =0 , Σ_(h=1) ^n a_h ^2 =1 find minimum value of Summation a_1 a_n +Σ_(h=1) ^(n−1) a_h a_(h+1) (korea university math contest problem)


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Question Number 213283 by issac last updated on 02/Nov/24
$a_h is Cauchy Sequence. Sequence {a_h }_(h=1) ^n Satisfy Σ_(h=1) ^n a_h =0 , Σ_(h=1) ^n a_h ^2 =1 find minimum value of Summation a_1 a_n +Σ_(h=1) ^(n−1) a_h a_(h+1) (korea university math contest problem)$
$a_{h} is Cauchy Sequence .$ $Sequence {a_{h}}_{h = 1}^{n} Satisfy \underset{h = 1}{\sum^{n}} a_{h} = 0, \underset{h = 1}{\sum^{n}} a_{h}^{2} = 1$ $find minimum value of Summation$ $a_{1} a_{n} + \underset{h = 1}{\sum^{n - 1}} a_{h} a_{h + 1}$ $(korea university math contest problem)$
	Answered by MrGaster last updated on 02/Nov/24

	$\underset{h = 1}{\sum^{n}} a_{h} = 0, \underset{h = 1}{\sum^{n}} a_{h}^{2} = 1$ $\underset{h = 1}{\sum^{n - 1}} a_{h} a_{h + 1} + a_{1} a_{n} = S$ ${(\underset{k = 1}{\sum^{n}} a_{h})}^{2} = (\underset{h = 1}{\sum^{n}} a_{h}) (\underset{k = 1}{\sum^{n}} a_{k}) = \underset{h = 1}{\sum^{n}} a_{h} a_{k} = 0$ $\underset{h = 1}{\sum^{n}} a_{h}^{2} + 2 \sum_{1 \leq h \leq k \leq n} a_{h} a_{k} = 0$ $1 + 2 \sum_{1 \leq h < h \leq n} a_{h} a_{k} = 0$ $\sum_{1 \leq h < h \leq n} a_{h} a_{k} = - \frac{1}{2}$ $S = \underset{h = 1}{\sum^{n - 1}} a_{h} a_{h} + 1 + a_{1} a_{n} = \sum_{1 \leq h < h \leq n} a_{h} a_{k} + a_{1} a_{n} - \sum_{1 < h < k \leq n_{k \neq h + 1}} a_{h} a_{k}$ $S = - \frac{1}{2} + a_{1} a_{n} - \sum_{1 \leq h < k \leq n_{k \neq h + 1}} a_{h} a_{k} - \underset{h = 1}{\sum^{n - 1}} a_{h} a_{h + 1} = - \frac{1}{2} - S$ $S = - \frac{1}{2} + a_{1} a_{n} + \frac{1}{2} + S$ $0 = a_{1} a_{n} + S$ $S = - a_{1} a_{n}$ $U sing C auchy - schwarz inequality :$ ${(\underset{h = 1}{\sum^{n}} a_{h} b_{h})}^{2} \leq (\underset{h = 1}{\sum^{n}} a_{h}^{2}) (\underset{h = 1}{\sum^{n}} b_{h}^{2})$ $S e t b_{n} = a_{h + 1}, b_{n} = a_{1}$ ${(\underset{k = 1}{\sum^{n}} a_{h} a_{h} + 1)}^{2} \leq (\underset{h = 1}{\sum^{n}} a_{h}^{2}) (\underset{h = 1}{\sum^{n}} a_{h + 1}^{2}) {(\underset{h = 1}{\sum^{n}} a_{h} a_{h + 1})}^{2} \leq 1$ $- \sqrt{1} \leq \underset{k = 1}{\sum^{n}} a_{h} a_{h + 1} \leq \sqrt{1}$ $- 1 \leq S \leq 1$ $T hus, the minimum value of S is \begin{matrix} - 1 \end{matrix} .$
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