Progressions. i i i 



Whence, substituting this vakie of ar" in (2) we obtain as an- 

 other expression for ^ 



rl—a 



r— I * ^^ 



10. To Insert any Number of Geometrical Means Be- 

 tween TWO Given Quantities. Suppose we are to insert /> 

 geometrical means between the two terms a and /. The whole 

 number of terms in the progression is therefore />-\-2. Hence, 

 substituting /-f^ for ?i in (i), Art. 8, 



Consequently 



and now, since the ratio is known, an}^ number of means can be 

 found by repeated multiplications. 



11. The two equations " • 



I af' — a 



V- r—i 



contain five different quantities. If any two of them are unknown, 

 and the values of the rest are given, the values of the two un- 

 known can be determined by a solution of the system. But if r 

 is an unknown quantit}^ the equations of the system are of a high 

 degree, since n is usually a large number and always greater than 

 2 at least. In this case we will be unable to solve the system, as 

 it is one beyond the range cf Chapter VII. Also if n is an un- 

 known quantity, we will have an equation with the unknown 

 quantity appearing as an expo7ie7it, which is a kind of equation 

 we have not yet discussed. Hence there are a limited number of 

 cases in which we can solve the above system. The following ta- 

 ble contains the ten possible cases, with the solution? as far as 

 possible. The values of n in the last four are printed merely to 

 make the table complete, for the manner of obtaining them is not 

 explained until Chapter XV is reached. 



