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 i 2 J- ( — ]) a 



2d 



3d 



4th 



u {g + l-(-l) S } (-1)* 



W {2+l-(-.l)* + l-(~l) a . l_-(_i) 6 J (_!)* 



5th 



( 2+l-(-l) +l-_(_l) .1_(-1) + i_ ( _ 1) a 

 .l-(-l)\ l-(-l) C } (-1)* 



y + i-(-i/+ i-(_i) c . i-(-i/+ H-i) *. i-(-i) c . i-(-i) 4 



/ a b — c d \ 



I + i-(-i) .i-(-i) .i-(-i) • l-c-i) } 



(-1) 



&c. &c. &c. 



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

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



2w | (— 1)" + (— 1) & +(— 1) C + ... &c} ; 



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

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

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

 plying 2 W, is p — 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 j but they are insufficient for its complete solu- 

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

 sion 



