# n-lines-are-drawn-inside-a-circle-in-such-a-way-that-the-circle-has-been-divided-in-maximum-number-of-parts-Determine-this-maximum-number-

Question Number 2642 by Rasheed Soomro last updated on 24/Nov/15
$${n}\:{lines}\:{are}\:{drawn}\:{inside}\:{a}\:{circle}\:{in}\:{such}\:{a}\:{way}\:{that}\: \\$$$${the}\:{circle}\:{has}\:{been}\:{divided}\:{in}\:{maximum}\:{number}\:{of} \\$$$${parts}.\:{Determine}\:{this}\:{maximum}\:{number}. \\$$
Commented by RasheedAhmad last updated on 24/Nov/15
$$\bullet{One}\:{line}\:{can}\:{divide}\:{the}\:{circle} \\$$$${atmost}\:\mathrm{2}\:{parts}. \\$$$$\bullet{Two}\:{lines}\:{can}\:{divide}\:{the}\:{circle} \\$$$${atmost}\:\mathrm{4}\:{parts}. \\$$$$\bullet{Three}\:{lines}\:{can}\:{divide}\:{the}\:{circle} \\$$$${in}\:{atmost}\:\mathrm{7}\:{parts}. \\$$$$…. \\$$$$… \\$$$$\:\bullet{n}\:{lines}\:{can}\:{divide}\:{the}\:{circle}\:{in} \\$$$${atmost}\:\left({say}\right)\:{m}\:{parts}. \\$$$$\frac{\mathcal{W}{hat}\:{is}\:{m}?}{} \\$$
Answered by prakash jain last updated on 24/Nov/15
$${n}\:\mathrm{lines}\:\mathrm{will}\:\mathrm{divide}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{maxium}\:\mathrm{number} \\$$$$\mathrm{of}\:\mathrm{parts}\:\mathrm{when} \\$$$$\bullet\:\mathrm{3}\:\mathrm{or}\:\mathrm{more}\:\mathrm{lines}\:\mathrm{are}\:\mathrm{not}\:\mathrm{concurrent} \\$$$$\bullet\:\mathrm{All}\:\mathrm{intersection}\:\mathrm{points}\:\mathrm{are}\:\mathrm{inside}\:\mathrm{the}\:\mathrm{circle} \\$$$$\mathrm{Let}\:{a}_{{i}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{parts}\:\mathrm{circle} \\$$$$\mathrm{is}\:\mathrm{divided}\:\mathrm{into}\:\mathrm{after}\:\mathrm{drawing}\:{i}^{{th}} \:\mathrm{line}. \\$$$${a}_{\mathrm{0}} =\mathrm{1}\:\left({no}\:{lines}\:{are}\:{drawn}\right) \\$$$${a}_{\mathrm{1}} ={a}_{\mathrm{0}} +\mathrm{1} \\$$$${a}_{\mathrm{2}} ={a}_{\mathrm{1}} +\mathrm{2} \\$$$${a}_{\mathrm{3}} ={a}_{\mathrm{2}} +\mathrm{3} \\$$$${a}_{\mathrm{4}} ={a}_{\mathrm{3}} +\mathrm{4} \\$$$$… \\$$$${a}_{{n}} ={a}_{{n}−\mathrm{1}} +{n} \\$$$${m}={a}_{{n}} =\mathrm{1}+\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\$$
Commented by Rasheed Soomro last updated on 25/Nov/15
$$\mathcal{E}{xcellent}\:\:\:\mathcal{S}{ir}! \\$$