If n people walk into a bar, the first asks for a pint of lager, the second asks

Question

If n people walk into a bar, the first asks for a pint of lager, the second asks for half a pint, the third for a quarter, etc... (Each person requesting half of what the former ordered). How many pints of lager will the bartender pour in total? How much money will you have if you deposit: $_n per month for a year, into a high-yield savings account, accruing m% annual interest, compounded monthly? How many distinct pizzas can be made with up to m topping? How many distinct anagrams of your surname are there? How many 5-card hands have a single pair and no 3-of-a-kind or 4-of- a-kind? How many 5-card hands have exactly a pair of aces?

Explanation / Answer

total pints pints of lager = 1pint +1/2 pint +1/4 pint +-----------

( geometric series with first term a=1 , common ratio r= 1/2

and sum of infinte term of a geometric series S = a/1-r )

Total= 1/ [ 1-1/2 ] = 1/ [ 1/2] = 2

b. Amount after 1 st month = n( 1+ m/1200)

Amount after 2nd month = [ n+n(1+m/1200) ] (1+m/1200)= n[ (1+m/1200)+(1+m/1200)2] and so on

   Amount after 12 months = n[ (1+m/1200) + (1+m/1200)2+------((1+m/1200)12] Let x= 1+m/1200

= n{ Sum of 12 terms ofGeometric progression with a= n , r= 1+m/1200 }

= n[x+x2+ -------x12] = n[ x( x12-1)/(x-1)] where x= 1+m/1200

  

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