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Question Number 196111 by sonukgindia last updated on 18/Aug/23

	Answered by mahdipoor last updated on 18/Aug/23
	$⇒⇒⇒⇒ if n≤x ⇒ Σ=(x−1)+...+(x−n)=nx−((n^2 +n)/2)= x^2 −(n+1)x+(((n^2 +2n+5)/4)) ⇒x^2 +(−2n−1)x+(((3n^2 +4n+5)/4))=0 ⇒x=n+((1+(√(n^2 −4)))/2)≥n (n≥2) {note:x≥n , x=n+((1−(√(n^2 −4)))/2)<n ⇒∄} ⇒⇒⇒⇒⇒ if x≤1 ⇒ ... ⇒x=((1−(√(n^2 −4)))/2)≤1 (n≥2) ⇒⇒⇒⇒⇒ if 1<x<n ⇒ get⌊x⌋=a Σ=[(x−1)+(x−2)+...+(x−⌊x⌋)]+ [(⌊x⌋+1−x)+....+(n−1−x)+(n−x)]= ((a(2x−1−a))/2)+(((n−a)(a+1+n−2x))/2)= ((n^2 +n)/2)−nx+2ax−a^2 −a=x^2 −(n+1)x+((n^2 +2n+5)/4) ⇒x^2 −(2a+1)x+(((5−n^2 )/4)+a^2 +a)=0 ⇒x=a+((1±(√(n^2 −4)))/2) ,⌊x⌋=a ⇒⌊a+((1±(√(n^2 −4)))/2)⌋=a ⌊((1±(√(n^2 −4)))/2)⌋=0⇒only when n=2⇒x=(3/2) ⇒⇒⇒⇒⇒⇒ a_1 =∄ ( get a_1 =0 !) a_2 =2.5+0.5+1.5=4.5 a_(n≥3) =n+((1+(√(n^2 −4)))/2)+((1−(√(n^2 −4)))/2)=n+1 ⇒⇒⇒⇒⇒⇒ 2(a_1 +...+a_(20) )=1.5+Σ_2 ^(20) (n+1)=229.5$
	$\Rightarrow\Rightarrow\Rightarrow\Rightarrow i f n ⩽ x \Rightarrow$ $Σ = (x - 1) + . . . + (x - n) = n x - \frac{n^{2} + n}{2} =$ $x^{2} - (n + 1) x + (\frac{n^{2} + 2 n + 5}{4})$ $\Rightarrow x^{2} + (- 2 n - 1) x + (\frac{3 n^{2} + 4 n + 5}{4}) = 0$ $\Rightarrow x = n + \frac{1 + \sqrt{n^{2} - 4}}{2} ⩾ n (n ⩾ 2)$ ${n o t e : x ⩾ n, x = n + \frac{1 - \sqrt{n^{2} - 4}}{2} < n \Rightarrow ∄}$ $\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow i f x ⩽ 1 \Rightarrow$ $. . .$ $\Rightarrow x = \frac{1 - \sqrt{n^{2} - 4}}{2} ⩽ 1 (n ⩾ 2)$ $\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow i f 1 < x < n \Rightarrow g e t ⌊ x ⌋ = a$ $Σ = [(x - 1) + (x - 2) + . . . + (x - ⌊ x ⌋)] +$ $[(⌊ x ⌋ + 1 - x) + . . . . + (n - 1 - x) + (n - x)] =$ $\frac{a (2 x - 1 - a)}{2} + \frac{(n - a) (a + 1 + n - 2 x)}{2} =$ $\frac{n^{2} + n}{2} - n x + 2 a x - a^{2} - a = x^{2} - (n + 1) x + \frac{n^{2} + 2 n + 5}{4}$ $\Rightarrow x^{2} - (2 a + 1) x + (\frac{5 - n^{2}}{4} + a^{2} + a) = 0$ $\Rightarrow x = a + \frac{1 \pm \sqrt{n^{2} - 4}}{2}, ⌊ x ⌋ = a \Rightarrow ⌊ a + \frac{1 \pm \sqrt{n^{2} - 4}}{2} ⌋ = a$ $⌊ \frac{1 \pm \sqrt{n^{2} - 4}}{2} ⌋ = 0 \Rightarrow o n l y w h e n n = 2 \Rightarrow x = \frac{3}{2}$ $\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow$ $a_{1} = ∄ (g e t a_{1} = 0!)$ $a_{2} = 2.5 + 0.5 + 1.5 = 4.5$ $a_{n ⩾ 3} = n + \frac{1 + \sqrt{n^{2} - 4}}{2} + \frac{1 - \sqrt{n^{2} - 4}}{2} = n + 1$ $\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow\Rightarrow$ $2 (a_{1} + . . . + a_{20}) = 1.5 + \underset{2}{\sum^{20}} (n + 1) = 229.5$
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