Ai/ii:r,RA. 29 



150. If a, 6, c be in n.r., then will 



1 4 111 



- + -- + -- r= --- . 



6 c ca ab c a 



151. If a, b, c, d be four positive quantities in n. r., 



a + d > b + c. 



152. Prove that 6 + c, c + a, a + b will be in n. p., if a f , 6", c* 

 be in A. p. 



153. If three numbers be in G. p. and the mean be added to 

 each of the three, the three sums will be in H. p. 



154. If n harmonic means be inserted between two positive 

 quantities a and 6, the difference between the first and last of 

 these means bears to the difference between a and b a ratio less 

 than n I : n + 1. 



155. If , a,, a 9 ... be an A. p., b , b l9 b t ... a O.P., and A, B % 

 C, D be any four consecutive terms of the series f& , , -^ 6,, 

 <* + *, > then will 



IX. Permutations and Combination*. 



The number of permutations of n different things taken r 

 together is denoted by m F r , and the corresponding number of com- 

 binations by m C f . 



. !':... " priori 

 ._,/>,+ 2r.. I />._,*r(r-l)._ 1 7>,., 1 



> + r(r-l)(r-2).../' 



:ig a whole number < r. 



