पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/३६७

CHAPTER VI-MIXED PROBLEMS. Examples in illustration thereof. 293. The sum of the series is 40; -the number of terms is 5 ;- and the common difference is 3; the first term is not known now. (Find it out.) When the first term 2, find out the common difference The rule for arriving at the sum and the number of terms in a series in arithmetical progression (with the aid of the known labha, which is the same as the quotient obtained by dividing the sum by the unknown number of terms therein) :- 294. The labha is diminished by the first term, and (then) divided by the half of the common difference; and on adding one to this same (resulting quantity), the number of terms in the series (is obtained). The number of terms in the series multi- plied by the labha becomes the sum of the series. An example in illustration thereof. 295. (There were a number of utpala flowers, representable as the sum of a series in arithmetical progression, whereof) 2 is the first term, and 3 the common difference. A number of women divided (these) utpala flowers (equally among them). Each woman had 8 for her share. How many were the women, and how mauy the flowers? The rule for arriving at the sum of the squares (of a given number of natural numbers beginning with one) :- 296. The given number is increased by one, and (then) squared ; (this squared quantity is) multiplied by two, and (then) diminished by the given quautity as increased by one. (The remainder 294 Algebraically, n=¹+1, where 1 = b 2 169 2 22 {2 (n+1)²-(n+¹) } // which is the labha. 295 The number of women in this problem is conceived to be equal to the number of terms in the series. 296 Algebraically, squares of the natural numbers up to n. 3 S₂, which is the sum of the 22