Six-free-hand-lines-are-drawn-inside-a-circle-in-order-to-divide-it-into-maximum-number-of-parts-Determine-this-number-Note-that-a-free-hand-line-has-following-three-properties-i-It-doesn-t-cut-

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Question Number 4222 by Rasheed Soomro last updated on 02/Jan/16

$$Sixfree-handlinesaredrawninside$$$$acircleinordertodivideitintomaximum$$$$numberofparts.Determinethisnumber.$$$$Notethatafree-handlinehasfollowing$$$$threeproperties:$$$$i)It{doesn}^{\prime }tcutitself.$$$$ii)Itjoinstwopointsofcircle.$$$$iii)Itcancutanotherlineatmostatthree$$$$points.$$$$Whatifthenumberoflinesisn?$$

Commented by prakash jain last updated on 03/Jan/16

$$a_0=1$$$$a_n=a_{n-1}+m(n-1)+1$$$$a_n=1+\frac{m(n-1)n}{2}+n$$

Commented by prakash jain last updated on 03/Jan/16

$$1st\mathrm{line}-2\mathrm{parts}$$$$2\mathrm{nd}\mathrm{line}-6\mathrm{parts}$$$$n^{th}\mathrm{line}-3(n-1)+1\mathrm{parts}\mathrm{added}$$$$a_0=1$$$$a_n=a_{n-1}+3n-2$$$$a_n=1+3\left(1\right)-2+3\left(2\right)-2+3\left(3\right)-2+\dots +3\left(n\right)-2$$$$a_n=1+\frac{3n^2+3n}{2}-2n$$$$a_n=\frac{2+3n^2+3n-4n}{2}=\frac{3n^2-n+2}{2}$$

Commented by Rasheed Soomro last updated on 03/Jan/16

$$\mathcal{TH}\alpha n{\mathbb{k}}^{\mathcal{S}}!$$$$Whatifthefree-handlinecancut$$$$otherlineatmostatmplaces.$$

Commented by Rasheed Soomro last updated on 03/Jan/16

$$\mathbb{T}\mathbf{h}\mathit{\alpha }\mathbf{n}\mathbb{X}\mathrm{again}!$$$$\mathbb{T}\text{boldsymbol}\mathrm{h}\alpha \mathrm{n}\mathbb{X}\mathrm{again}!$$

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