We shall for convenience of proof regard a1 as being positive. If r has a numerical value less than unity, the absolute value of a and accordingly of a1r"/(1 − r) will decrease as n increases in value. Accordingly, by giving to n a value great enough, we may make the value of a1"/(1 − r) as small as we please, but we can never in this way make the value of the fraction zero. − r), As the value of the fraction a"/(1 − r) diminishes, the sum S approaches more nearly the value of the first fraction a1/(1 but it never becomes exactly equal to it, because α1”/(1 − r) can never become zero. We may, by taking n great enough, make the sum become and remain as nearly equal to a1/(1 -r) as we please. This is expressed by saying "the limit of the sum of an infinite number of terms of a decreasing geometric progression is a1/(1 − r).” Expressed in symbols, we have: lim S = 1 -r which is read, "the limit of the sum of an infinite number of terms (of a given geometric progression) is equal to a1/(1 − r).” As an alternative form we have So = 01 1 r' which is read, "the sum of an infinite number of terms (of a given geometric pro Ex. 1. Find the sum of an infinite number of terms of the decreasing geometric progression 1, 1/2, 1/4, 1/8, Substituting the value 1 for a1, and 1/2 for r, in the formula Ex. 2. Find the sum of an infinite number of terms of 36, 12, 4, - 51. By means of the formula for the sum of an infinite number of terms of a decreasing geometric progression we may obtain the generating fraction of a repeating decimal fraction, that is, the fraction which gives rise to a repeating decimal fraction if the numerator is divided by the denominator. Ex. 3. Find the generating fraction of the repeating decimal fraction .3. It should be observed that the dot written above the 3 indicates that 3 is to be repeated indefinitely; that is, .3 = .333333 +. We may write .3333 ・ ・ ・ 1% +180 + 1800 + 10800 + In this form the repeating decimal fraction appears as a decreasing geometric series the first term of which is 3/10, and the common ratio of which is 1/10. Ex. 4. Find the improper fraction which may be transformed into the repeating decimal fraction 3.236. We may write 3.236 = 3.2 + .036. It should be understood that the dots above the 3 and 6 denote that 36 is to be repeated indefinitely, that is, .036 = .0363636363636 + · The required fraction may be obtained by finding the sum of 3.2 and 2/55, which is found to be 178/55. The student should show, by dividing the numerator by the denominator, that the fraction 178/55 gives rise to the given repeating decimal fraction 3.23636+. 52. The process of finding the generating fraction corresponding to any given repeating decimal fraction is sometimes spoken of as evaluating the given repeating decimal fraction. EXERCISE XXVII. 8 Find the sum of an infinite number of terms of each of the following series : 19. Find r, having given a, 6 and a, = 384. 20. Find two numbers the difference of which is 48 and the geometric mean of which is 7. 21. Find So, having given a 109 = 72 and a = 64/3. 1/968. 22. Find a1, provided that a = 1/32 and a ̧ 23. Find a,, knowing that a, = .008 and a = .000064. 24. The difference between two numbers is metic mean exceeds their geometric mean by 25. 70 and their arithFind the numbers. 25. Find three numbers in geometric progression such that their sum shall be 14 and the sum of their squares 84. 26. The sum of the first four terms of a geometric progression is 15, and the sum of the next two terms is 48. Find the progression. 27. A number consists of three figures in geometric progression. The sum of the figures is 7, and if 297 be added to the number the order of the figures will be reversed. Find the number. 28. A ball is thrown vertically upward to a height of 120 feet and after falling it rebounds one-third of the distance, and so on. Find the whole distance passed over by the ball before it comes to rest. MENTAL EXERCISE XXVII. 9 Classity each of the following as Arithmetic, Harmonic, or Geometric Progressions: EXERCISE XXVII. 10. Review Simplify each of the following: a 1 4. +1 1 + a a 1 1 ()() 5. (a−2 — b−2) ÷ (a − b). a a1+b-1 b-1 6. a1 + b-1 8. If a 1, b2, c=-3, find the value of 9. Simplify [(x + y)√x − y][ (x − y)√x + y]. 10. Simplify (V2 — y2)(√x + y)(√x − y). |