32 BOOK OF MATHEMATICAL PROBLEMS. 



168. The sum of the first r+l coefficients of the expansion 



\m + r 



of (1 - x) m is equal to ' . 



rnr 



169. Prove that 



, 



(f-3)(n-4)(n-g) 



[3 (l + m) a m-l 



(^ _ 3) (n - 4) (w - 5) . 



1 -* + 



n being a positive integer. 



170. If p be nearly equal to q, then will - ~ be nearly 

 equal to 



171. If a r denote the coefficient of x r in the expansion of 



( _t_? J in a series of ascending powers of x; the following rela- 

 \1 x/ 



tion will hold among any three consecutive coefficients, 

 (r + 1) a r+l - 2na r -(r-l) a r _, = 0. 



/I _l_ /*\" 



172. If yr^ -^ be expanded in ascending powers of x, the 



coefficient of x n+r ~ l is (n + 2r) 2"" 1 , w, r being positive integers 

 (including zero). 



