158 AN EXAMINATION OF SOME QUESTIONS 



We can now represent the profit or loss on each event, with- 

 out determining which of the two it is : 



On the 1st he gams u \ 3 ^ ( — 1)° 



2d u |2 + 1_(— 1)"} (_1)* 



3d „ |2 + l_(_l/ + l-{-l)°- l-(-l)*} (-!)• 



4th 



5th 



|2+l-(-l)' +l-(-l) .l-(-l) +1_(_1)'' 

 .l-(-l)'. l_(_l/} (-1)" 



ig + l-C-l)^ l-(-l) ' . !-(-!)''+ l-{-l)*. l-(-l) ' ■ l-(-l)' 



(. +i-(-i) .i_(-i) .i-(-i) -M-i) ] .(-1) 



&c. &c. &c. 



The sum of all these is the object sought : with respect to that 

 part which multiplies 2 u, it is easily found ; it is 



2« |(-l)'' + (-l)^+(-ir + ...&c.}; 



but we know that he has gained p, and lost q times ; conse- 

 quently p of the quantities a, b, c, &c. are even numbers, and 

 q of them odd ones; therefore the sum of the series, multi- 

 plying 2 m, is j9 — q, and the value of the part alluded to is 



2{p — q)u. 

 So far, then, we can proceed by means of the data furnished in 

 the question ; but they are insufficient for its complete solu- 

 tion, on attempting to find the sum of that part of the expres- 

 sion 



