Question 1187468
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You are running a foundation for street children. Mr. Park, a rich benefactor, offers two options for his donations :
OPTION A: To give $ 1,000 on day 1, $999 on day 2, $998 on day 3, with the process to end after 1,000 days.
OPTION B: To give $1 on day 1, $2 on day 2, $4 on day 3, for 19 days.
You have to tell him today which option you want to take.
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OPTION (A) :  You need to find the sum of the first 1000 natural numbers

                  {{{S[A]}}} = 1000 + 999 + 998 + . . . + 2 + 1.


              In your high school, you diligently learned about arithmetic progression, and you know

              that the sum of the first n natural numbers is  {{{(n*(n+1))/2}}}.

              So, you have a happy opportunity to apply your knowledge, and you quickly calculate


                  {{{S[A]}}} = {{{(1000*1001)/2}}} = 500*1001 = 500500 dollars.



OPTION (B) :  You need to find the sum of the first 19 terms of the geometric progression
              with the first term of 1 and the common ratio of 2.

                  {{{S[B]}}} = {{{1}}} + {{{2}}} + {{{2^2}}} + . . . {{{2^18}}}.


              In your high school, you diligently learned about geometric progression, and you know

              that the sum of the first n terms of the given progression is  {{{2^19-1}}}.

              So, you have a happy opportunity to apply your knowledge, and you quickly calculate


                  {{{S[B]}}} =  {{{2^19 - 1}}} = 524287 dollars.


Now you have everything to make your decision and announce it to Mr. Park.
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I am sure you will make a right decision, &nbsp;and the street children will be happy &nbsp;(&nbsp;!&nbsp;)


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Solved.