#### For what value of n are the nth terms of two A.Ps 63 , 56 , 67 , ....... and 3 , 10 , 17 .......equal? =============================================== The diameter of a roller 120cm long is 84cm . If it takes 500 complete revolutions to level a playground , determine the cost of levelling it at the rate of 30 paise per square metre.

$1)\;63,65,67,......\\* a=63,\;d=2\\* a_{n}=a+(n-1)d\\*\Rightarrow a_{n}=63+ (n-1)(2)=61+2n\\*3,10,17\;....\\*a=3,\;d=7\\*a_{n}=a+(n-1)d\\*\Rightarrow a_{n}=7n-4\\* according\;to\;question\;they\;are\;equal\\* \Rightarrow 61+2n=7n-4\\* \Rightarrow 65=5n\Rightarrow n=13\\*So,\;for\;n=13\;a_{13}\;of\;both\;AP\;are\;equal\\* 2)\;The\;roller\;is\; a\;right\;circular\;cylinder\;of\;height,h=120\;cm\\* and,\;radius\;of\;its\;base,r=42\;cm\\*\therefore area\;covered\;by\;the\; roller\;in\;one\;revolution=curved\;surface\;area\;of\;the\;roller\\*\Rightarrow 2\times \pi\times 42\times120=31667.25\;cm^2\\*so,\;area\; covered\;by \;the\;roller\;is\;500\;revolution=(31667.25\times500)cm^2\\* \Rightarrow \frac{31667.25\times 500}{100\times 100}\;m^2=1583.36\;m^2\\* cost\;of\;levelling\;the\;playground=1583.36\frac{30}{100}=475\\* Hence,\;cost\;of\;levelling\;the\;playground\;is\;Rs\;475$