ALGEBRA. 



137. The radix of a scale in which 49 denotes a sq 

 number must be of the form (r + 1) (r + 4), where r is some \ 

 number. 



138. The radix of a scale being 4r+2, prove that if the 

 iii the units' place of any number be either 2r + 1, or 2r + 2, 



the square of the number will have the same digit in the units' 

 place. 



139. Find a number of (1) three digits, (2) four digits, in the 

 denary scale such that if the first and last digits be interchanged, 

 the result represents the winie number in the nonary scale, and 

 prove that there is only one solution in each case. 



140. If the radix of any scale have more than one prime 

 factor, th.- re will exist at least one digit different from unity such 

 that if any number have that digit in the units' place, its square 

 will have the same digit in the units' place. 



V 1 1 1. A . ''utietical, Geometrical, and Uarmonical Progressions. 



L If the sum of m terms of an A. p. be to the sum of // 

 terms as mf : n*; prove that the m* term will be to the n* 

 term as 2m -1 : 2*1-1. 



The series of natural numbers are divided int.. gr 



, 8, 9 ; and so on : hat the sum of the 



.i.t:r in the n* group is n*+ (- I) 1 . 



143. The sum of the products of every two of n terms t>: 

 whose first term is a and last term I, is 



