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Question Number 64903 by naka3546 last updated on 23/Jul/19

$$1,3,7,15,30,57,103,x$$$${What}^{\prime }sx?$$

Commented by Tony Lin last updated on 23/Jul/19

$$f\left(1\right)=1,f\left(2\right)=3,f\left(3\right)=7,f\left(4\right)=15$$$$f\left(5\right)=30,f\left(6\right)=57,f\left(7\right)=103,f\left(8\right)=x$$$$f\left(x\right)$$$$=f\left(1\right)\frac{(x-2)(x-3)(x-4)(x-5)(x-6)(x-7)}{(1-2)(1-3)(1-4)(1-5)(1-6)(1-7)}$$$$+f\left(2\right)\frac{(x-1)(x-3)(x-4)(x-5)(x-6)(x-7)}{(2-1)(2-3)(2-4)(2-5)(2-6)(2-7)}$$$$+f\left(3\right)\frac{(x-1)(x-2)(x-4)(x-5)(x-6)(x-7)}{(3-1)(3-2)(3-4)(3-5)(3-6)(3-7)}$$$$+f\left(4\right)\frac{(x-1)(x-2)(x-3)(x-5)(x-6)(x-7)}{(4-1)(4-2)(4-3)(4-5)(4-6)(4-7)}$$$$+f\left(5\right)\frac{(x-1)(x-2)(x-3)(x-4)(x-6)(x-7)}{(5-1)(5-2)(5-3)(5-4)(5-6)(5-7)}$$$$+f\left(6\right)\frac{(x-1)(x-2)(x-3)(x-4)(x-5)(x-7)}{(6-1)(6-2)(6-3)(6-4)(6-5)(6-7)}$$$$+f\left(7\right)\frac{(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)}{(7-1)(7-2)(7-3)(7-4)(7-5)(7-6)}$$$$x$$$$=f\left(8\right)$$$$=1\times \frac{6\times 5\times 4\times 3\times 2\times 1}{(-1)(-2)(-3)(-4)(-5)(-6)}$$$$+3\times \frac{7\times 5\times 4\times 3\times 2\times 1}{1\times (-1)(-2)(-3)(-4)(-5)}$$$$+7\times \frac{7\times 6\times 4\times 3\times 2\times 1}{2\times 1\times (-1)(-2)(-3)(-4)}$$$$+15\times \frac{7\times 6\times 5\times 3\times 2\times 1}{3\times 2\times 1\times (-1)(-2)(-3)}$$$$+30\times \frac{7\times 6\times 5\times 4\times 2\times 1}{4\times 3\times 2\times 1\times (-1)(-2)}$$$$+57\times \frac{7\times 6\times 5\times 4\times 3\times 1}{5\times 4\times 3\times 2\times 1\times (-1)}$$$$+103\times \frac{7\times 6\times 5\times 4\times 3\times 2}{6\times 5\times 4\times 3\times 2\times 1}$$$$=1-21+147-525+1050-1197+721$$$$=176$$

Commented by MJS last updated on 23/Jul/19

$$\mathrm{generally}\mathrm{it}\mathrm{could}\mathrm{be}\mathrm{any}\mathrm{number}.$$$${\mathrm{we}}^{\prime }\mathrm{re}\mathrm{interpreting}\mathrm{it}\mathrm{as}$$$$y=c_6{x}^6+c_5{x}^5+c_4{x}^4+c_3{x}^3+c_2{x}^2+c_{1}x+c_0$$$$\mathrm{with}\mathrm{given}\mathrm{pairs}(x,y):$$$$(1,1)(2,3)(3,7)(4,15)(5,30)(6,57)(7,103)$$$$$$$$\mathrm{but}\mathrm{we}\mathrm{can}\mathrm{use}\mathrm{any}\mathrm{function}\mathrm{with}7\mathrm{unknown}$$$$\mathrm{constants}$$$$y=\frac{1}{c_6{x}^6+c_5{x}^5+c_4{x}^4+c_3{x}^3+c_2{x}^2+c_{1}x+c_0}$$$$\mathrm{leads}\mathrm{to}\frac{3914}{781}$$$$y=\frac{c_6}{x^6}+\frac{c_5}{x^5}+\frac{c_4}{x^4}+\frac{c_3}{x^3}+\frac{c_2}{x^2}+\frac{c_1}{x}+c_0$$$$\mathrm{leads}\mathrm{to}\frac{43339367}{262144}$$$$y=\frac{1}{\frac{c_6}{x^6}+\frac{c_5}{x^5}+\frac{c_4}{x^4}+\frac{c_3}{x^3}+\frac{c_2}{x^2}+\frac{c_1}{x}+c_0}$$$$\mathrm{leads}\mathrm{to}\frac{1026031616}{5937579}$$$$\mathrm{we}\mathrm{could}\mathrm{also}\mathrm{use}$$$$y=c_7{x}^7+c_6{x}^6+c_5{x}^5+c_4{x}^4+c_3{x}^3+c_2{x}^2+c_{1}x+c_0$$$$\mathrm{then}\mathrm{we}\mathrm{have}\mathrm{one}\mathrm{factor}\mathrm{free}\mathrm{to}\mathrm{choose}:$$$$y=-\frac{c_0+2}{5040}{x}^7+\frac{4c_0+7}{720}{x}^6-\frac{46c_0+65}{720}{x}^5+\frac{56c_0+65}{144}{x}^4-\frac{967c_0+749}{720}{x}^3+\frac{469c_0+277}{180}{x}^2-\frac{11(99c_0-5)}{420}x+c_0$$$$x=8$$$$y=174-c_0$$$$\Rightarrow \mathrm{any}\mathrm{value}\mathrm{fits}$$

Answered by MJS last updated on 23/Jul/19

$$137153057103176$$$$24815274673$$$$247121927$$$$23578$$$$1221$$$$10-1$$$$-1-1$$

Commented by Tony Lin last updated on 23/Jul/19

$$Whatisthemethod?$$

Commented by MJS last updated on 23/Jul/19

$$abcd...$$$$b-ac-bd-c...$$$$(c-b)-(b-a)(d-c)-(c-b)...$$$$...$$$$\mathrm{until}\mathrm{you}\mathrm{reach}\mathrm{a}\mathrm{line}\mathrm{with}\mathrm{each}\mathrm{number}\mathrm{of}$$$$\mathrm{the}\mathrm{same}\mathrm{value}.\mathrm{then}\mathrm{start}\mathrm{at}\mathrm{the}\mathrm{bottom}$$$$\mathrm{adding}\mathrm{this}\mathrm{number}\mathrm{once}\mathrm{more}\mathrm{and}\mathrm{go}\mathrm{up}$$$$\mathrm{from}\mathrm{line}\mathrm{to}\mathrm{line}.\mathrm{this}\mathrm{gives}\mathrm{the}\mathrm{same}\mathrm{result}$$$$\mathrm{as}\mathrm{the}\mathrm{polynomial}\mathrm{approach}$$$$\mathrm{we}\mathrm{can}\mathrm{say}$$$$\mathrm{the}{1}^{\mathrm{st}}\mathrm{line}\mathrm{is}y$$$$\mathrm{the}{2}^{\mathrm{nd}}\mathrm{line}\mathrm{is}△y$$$$\mathrm{the}{3}^{\mathrm{rd}}\mathrm{line}\mathrm{is}{△}^{2}y$$$$\mathrm{the}{n}^{\mathrm{th}}\mathrm{line}\mathrm{is}{△}^{n-1}y$$$${\mathrm{it}}^{\prime }\mathrm{s}\mathrm{similar}\mathrm{to}\mathrm{the}{n}^{\mathrm{th}}\mathrm{derivate}\mathrm{of}\mathrm{a}\mathrm{polynome}$$$$\mathrm{of}\mathrm{degree}k.\mathrm{you}\mathrm{reach}\mathrm{the}\mathrm{constant}\mathrm{factor}$$$$\mathrm{after}k\mathrm{steps}$$

Commented by Tony Lin last updated on 23/Jul/19

$$isthemethodonlybeusedon$$$$conditionthata,b,c,dareinA.P?$$

Commented by MJS last updated on 23/Jul/19

$$\mathrm{no}.\mathrm{write}\mathrm{down}\mathrm{any}\mathrm{numbers},\mathrm{it}\mathrm{always}\mathrm{works}$$$$\mathrm{but}{\mathrm{you}}^{\prime }\mathrm{ll}\mathrm{only}\mathrm{find}\mathrm{polynomial}\mathrm{solutions}$$$$1123581329$$$$01123516$$$$1011211$$$$-11019$$$$2-118$$$$-327$$$$55$$

Commented by Tony Lin last updated on 23/Jul/19

$$thankssir.igotit.$$

Commented by MJS last updated on 23/Jul/19

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{welcome}$$

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