Examples in Progression. 4. Find n and S, when a, l, and r are known. i a r 5. Find the number and sum of terms of the series of which the first term is 6, the last term 796, and the common difference 10. Ans. The number of terms = 80, the sum = 32080. 6. Find r, when a, l, and n are known. 7. Find the common difference and sum of the series, of which the first term is 75, and the last term 15, and the number of terms 13. Ans. The common difference - 5, the sum = 585. 8. Find r and n, when a, l, and S are known. 28 Ans. n = ati' 12 28-(a+7) 9. Find the common difference and number of terms of a series, of which the first term is 2, and the last term 345, and the sum 8675. Ans. The number of terms = 50, the common difference = 7. 10. Find I and n, when a, r, and S are known. V[2r8+(a— }r)?] –(a– ļr), Ans. n = l=V[2r8+(a-1r)?] 1r. Examples in Progression. 11. Find the last term and number of terms of a series, of which the first term is 3, the common difference 4, and the sum of the terms 105. Ans. The last term = 27, the number of terms = 7. 12. Find a and n, when l, r, and S are known. 1+fr FV[(?+ 1r)2—2r8] Ans. n = a=#v[(i+1r)? — 2r8]+ fr. 13. Find the first term and the number of terms of a series, of which the last term is 13, the common difference 3, and the sum of the series 35. Ans. The first term = the number of terms = 5. -1, 14. Find I and r, when a, n, and S are known. 28 Ans. I= a, n an) n(n-1) 15. Find the last term and common difference of a series, of which the first term is $, the number of terms 12, and the sum 100. Ans. The last term = 16, the common difference = 11. Examples in Progression. 17. Find the first term and common difference of a series, of which the last term is 50, the number of terms 20, and the sum 600. Ans. The first term = 10, the common difference = 275. 18. Find a and S, when l, r, and n are known. Ans. a=1-(n-1) r, S= } [21—(n-1) r] n. 19. Find the first term and sum of the terms of a series, of which the last term is 100, the common difference 3, and the number of terms 51, Ans. The first term = 75, the sum of the terms = 44621. 20. Find a and I, when r, n, and 8 are known. s Ans. Q = - (n-1), s ++ r 21. Find the first and last terms of a series, of which the common difference is 5, the number of terms 6, and the sum 321. Ans. The first term = 41, the last term = 66. 22. Find the sum of the natural series of numbers 1, 2, 3, &c. up to n terms. Ans. f n (n + 1). 23. Find the sum of the natural series of numbers from 1 to 100. Ans. 5050. 24. Find the sum of the odd numbers 1, 3, 5, &c. up to in terms. Ans. n2. Examples in Progression. 25. Find the sum of the odd numbers from 1 to 99. Ans. 2500. &c. up 26. Find the sum of the even numbers 2, 4, 6, to n terms. Ans. n (n + 1). one. 27. Find the sum of the even numbers from 2 to 100. Ans. 2550. 28. One hundred stones being placed on the ground, in a straight line, at a distance of 2 yards from each other ; how far will a person travel, who shall bring them one by one to a basket, placed at 2 yards from the first stone? Ans. 11 miles, 840 yards. 29. We know, from natural philosophy, that, a body which falls in a vacuum, passes, in the first second of its fall, through a space of 167, feet, but in each succeeding second, 321 feet more than in the immediately preceding Now, if a body has been falling 20 seconds, how many feet will it have fallen the last second ? and how many in the whole time? Ans. 6274 feet in the last second, and 64333 feet in the whole time. 30. In a foundery, a person saw 15 rows of cannon-balls placed one above another, and asked a bombardier how many balls there were in the lowest row. easily calculate that,” answered the bombardier. “ In all these rows together, there are 4200 balls, and each row, from the first to the last, contains 20 balls less than the one immediately below it.” How many balls, therefore, were there in the lowest row ? Ans. 420. “ You may 252. The arithmetical mean between several Arithmetical Mean. quantities is the quotient of their sum divided by their number. Thus the arithmetical mean between the two quantities a and b is half their sum, or ) (a+b); that between the four quantities 1, 7, 11, 5 is 6. 253. Problem. To find the arithmetical mean between the terms of an arithmetical progression. Scholium. It is, by the preceding definition S or, since S = 1 n (a + 1), it is i (a + 1; that is, half the sum of the extremes, and also, by art. 248, half the sum of any two terms at equal distances from the extremes. 254. Problem. To find the first and last terms of a progression of which the arithmetical mean, the number of terms, and the common difference are known. Solution. If we denote the arithmetical mean by M, we have S ; M= which, substituted in the result of example 20, in art. 251, gives a= M - * (n − 1)r, 1= m + 1 (n − 1) r. 255. Scholium. In very many of the problems involving arithmetical progression, it is convenient |