414 FROM ELEMENTARY ALGEIiRA TO THE CALCULUS. 



I.e., a product is increased if tlie factors are replaced by two 

 which leave the sum unchanged, but are more nearly equal. 



Now consider ah c . . . z, the product of n quantities. 



Let jjt be their arithmetic mean, so that a + h+ . . + z = nfj. 

 Choose two of the quantities (a. z), so that a is greater than ^, 

 and z less than fx (this is always possible, of course). 



Then az <^i {a + z- n)—c2il\ the latter factor z' . 



Therefore ahc . . . z < ja. he . . . z , where h + c+ . . + z' 



= (w- i) /u., and .-. fj. is the arithmetic mean of &, c, z' . 



Make a similar change in the product he . . . z ; and, continuing 

 the process, we get finally ahc . . . z<n". 



or {ahc ... z)''' < -{a + h+ . .+z). 

 m index 



(ii) Take for the quantities ahc . . . z, i quantities each i, and 

 the remaining / equal to (i + .r), where x has any + value making 



[1+ x)+ . 



i + j (i + x) 

 Then (i + a;V"-+^' < — /rrv — - 



Therefore (i + .v)" < i + nx if w is a+ fraction. 



Thence follows {i + nx)'" > i + nx/n. 



or, in other words, (Ht.t)" >i + nx, if w >i. 



I 

 Now I +x =1 say. 



n 

 Therefore (i - x)"" > i + nx^i -n + n (i + x) =i - w + — — — 



.-. (i - x) ' " > (i - n) (i - x) + w = i- (i - w)x. 



But, n being > i here, (i - w) = any negative quantity 

 .-. generally {i-\- x)" > i + 7ix when n is - ''" , 



And, finally, (i + x)" < i + nx, according as x lies between and i. 

 or does not. 



Again, if {i + x)" ^ i+nx, which = (i + .r) + (« - i) x, 

 Dividing by (i + x)" , i ^ (i + .v)' "' + (n - i) x (i + x) ' , 

 therefore (i + xf^" >■ i + (i - w) ^ (i + -y) " . 

 But (i — n) lies or does not hv between and i exactly as n does, 



J.. r / \„ T 1 X I I + w,T I for all possible 



therefore ( I + .r) lies between , , , v„_i ^, ^ ^r ^ 



^ ' I i + nx{i + X) ' I values of w and a:, 



{i + nx) being the upper limit if n lies between and i, but the 



lower limit if n lies outside 0, i. 



We may write the result in the form —, < „, 



■' n{a - 1 I « 



/y' § (l'^' T /7^ T \ T 



whence, writing a' for a and - for n, \ - , - -^ ) < 



'^ X ^ X X -^ \ <^ 



Whence follows the reasoning by which on a previous page we 

 established the existence and properties of ' the logarithm,' 

 and we have D (a^) = rt'' log a. 



,/) -1 



