! !" " # $# !# %&' ( &)%)* # + &,* -, * %, * %$$,,( + * ( + * .&* /+ .&* /+ 0 01%%1%%+ + 1!234,5&* 6/&&%0.* * %0* &"7 # 8 9 && * &/00* &"&%0.* !,1342 # 8 1&:%%% 22443311,,!!1&%%* 22443311,,!!22443311,,!!( )$( $( ; . # 8 ( &< # ' * /8 $,' 1+ + 9)' )= .>= 6/+ &&+ && * + /&* 0 >( &)= ./ %%% "6/&+ &* ?&1 9?&. %?&. % * * %* /%* /%* /%* /1%1* /1%1* /!%!* /!%!* /77@ * >)6 &-< >)0* & 0&* &+ * 0A%&* 00%0* 0& * 0&0 * 5+ + &%%&&&071 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S21)(n n S+=1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S"6& * < Algebraic argument1 2 . . . . . . . . n= number of white dots.1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn (n+1) = 2S1 + 2 + 3 + . . . + n-1 + n = Sn + n-1 + n-2 + . . . + 2 + 1 = S(n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2Sn
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