view puzzle related to this thread.
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It is obvious that he is in East-North direction.
Distance travelled in North and South directions
= 1/2 - 1/8 + 1/32 - 1/128 + 1/512 - 1/2048 + and so on... (an ifinite GP with r = (-1/4) )
(1/2) * ( 1 - (-1/4)n )
= ---------------------------
( 1 - (-1/4) )
= 1 / ( 2 * ( 1 - (-1/4) ) )
= 2/5
Similarly in East and West directions
= 1 - 1/4 + 1/16 - 1/64 + 1/256 - and so on... (an infinite GP with r = (-1/4) )
(1) * ( 1 - (-1/4)n )
= ---------------------------
( 1 - (-1/4) )
= 1 / ( ( 1- (-1/4) )
= 4/5
So the soldier is 4/5 miles away towards East and 2/5 miles away towards North.Using pythagorus theorem

distance = (4/25+16/25)^0.5 =  (20/25^0.5 = 2 root 5 / 5 units towards N-E

:)