Math Question and Answers


Question and Answers Forum

All Questions Topic List
Arithmetic Questions
Previous in All Question Next in All Question
Previous in Arithmetic Next in Arithmetic
Question Number 82243 by M±th+et£s last updated on 19/Feb/20

	Answered by mind is power last updated on 19/Feb/20

	$o b s e r v a t i o n$ $\forall k \in N, \forall x \in R_{+}$ $1 . . . .$ $x^{k} - k x + k - 1 ⩾ 0$ $f (x) = x^{k} - k x + k - 1$ $f^{'} (x) = 0 \Leftrightarrow x = 1$ $f (1) = 0 ⩾ 0 \Rightarrow f (x) ⩾ 0, \forall x \in R_{+}$ $(\begin{matrix} n \\ 2 \end{matrix}) = \frac{n}{2} (n - 1) = \underset{k = 1}{\sum^{n}} (k - 1)$ $\underset{i = 1}{\sum^{n}} {i x}_{i} ⩽ (\begin{matrix} n \\ 2 \end{matrix}) + \underset{i = 1}{\sum^{n}} x_{i}^{i} \Leftrightarrow 0 ⩽ \underset{i = 1}{\sum^{n}} (i - 1) + \underset{i = 1}{\sum^{n}} x_{i}^{i} - \underset{i = 1}{\sum^{n}} {i x}_{i}$ $\Leftrightarrow 0 ⩽ \underset{i = 1}{\sum^{n}} (x_{i}^{i} - {i x}_{i} + i - 1)$ $b y 1 x_{i}^{i} - {i x}_{i} + i - 1 ⩾ 0$ $\Rightarrow \underset{i = 1}{\sum^{n}} (x_{i}^{i} - {i x}_{i} + i - 1) ⩾ \underset{i = 1}{\sum^{n}} (0) = 0$ $\Leftrightarrow \underset{i = 1}{\sum^{n}} {i x}_{i} ⩽ \underset{i = 1}{\sum^{n}} x_{i}^{i} + (\begin{matrix} n \\ 2 \end{matrix})$
	Commented by M±th+et£s last updated on 19/Feb/20

	$n i c e s o l u t i o n s i r t h a n k y o u$
	Commented by mind is power last updated on 20/Feb/20

	$w i t h e p l e a s u r$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum