CHAPTER III-FRACTIONS. three, the first and the last denominators so obtained being (however) multiplied (again by 2 and respectively. Examples in illustration thereof. 76. The sum of five or six or seven (different fractional) quantities, having 1 for (each of) their namerators, is 1 (in each case) O you, who know arithmetic say what the required) denominators are. The rule for finding out the denominators in the case of an nuoven number of fractions) 77. When the sum of the (different fractional, quantities, having une for each of their numerators, is one, the required) denominators are such as, beginning with two, go on successively rising in value by one, each (such denominator) being (further) multiplied by that 1 1 37-1 as! - the last fraction + '1 Fio. is it is ch that, when the fust fraction and 1 are added to tlus last result, the sum becomes 1. 3. 3" - 1 In this connection it may be noted that, in a series in geometrical progression consisting of n terms, having as the first term and as the contuon ratio, the 1 1 a a sum is, for all positive integral values of a, less than (1 2 x 3 x + ] 3 xx 1 1 2 x 33 x + = 2 { 2 ² ² = 2 { (+- -) + (-) + (-²) + = 2y = 1. (n + 1)th term in the series. Therefore, if we add to the sum of the series in geometrical progression I a - x the (n+1th term, which is the last fraction J a-1 1 4x5 + ... 1 - 1' 2 a according to the rule stated in this stanza. we get have to add in order to get 1 as the sam. This rule as the first fraction, and so 3 is the value chosen for a, since the numeraton of all the fractions has to be 1. autioned in the 77. Here note 1 "X by: 5, ! 1 1 n (x - 1) + 55 1 +...+ To this ]
76 } X the + 1 ], we 1 (n-1) 1 x "
