SECTION X. Examples of the Solution of Problems in Arithmetical and Geometrical Progressions. 1. A PERSON bought 7 books, the particular prices of which (in shillings) were in arithmetical progression. The price of the next above the cheapest was 8 shillings, and the price of the dearest 23 shillings. What was the price of each book? Let the price of the cheapest, and y = the common difference; then x + y = the price of the second = 8, and the prices are 5, 8, 11, 14, 17, 20, 23 shillings, respectively. A number consists of 3 digits, which are in arithmetical progression; and this number divided by the sum of its digits is equal to 26; but if 198 be added to it, the digits will be inverted. Required the number. then the number will be 100 (x − y) + 10x + x + y = and x = 3y. 99y198 = 100. (x + y) + 10x + (x − y) ..the digits are 2, 3, and 4, and the number = 234. = 3. The sum of £1. 78, was to be raised by subscription by three persons A, B, and C; the sums to be subscribed by them respectively forming an arithmetical progression. But C dying before the money was paid, the whole fell to A and B; and C's share was raised between them in the proportion of 3: 2, when it appeared that the whole sum subscribed by A was to the whole sum subscribed by B 4 5. Required the original subscriptions of A, B, and C. Let xy, x, x + y, be the respective subscriptions of A, B, and C; Now 5 2 then 3x27; and .. x = 9. (C's share =) 9+y: the part paid by B and 5: 3 :: 9+y: the part paid by A. (9 + y), 3 = 3 and consequently, A paid upon the whole 9 −y + 2 . (9 + y) .. the sums to be subscribed originally were 3, 9, and 15 shillings. 4. Four numbers are in arithmetical progression. The sum of their squares is equal to 276, and the sum of the numbers themselves is equal to 32. What are the numbers? Let 2y the common difference, also the sum of their squares = 4x2 + 20y2 = 276, in which substituting the value of a found above, 256 +20y2 = 276; by transposition, 2012 = 20; .. y2 = 1; 5. The sum of the squares of the extremes of four numbers in arithmetical progression is 200, and the sum of the squares of the means is 136. What are the numbers? Supposing as before, x+3y, x + y, xy, and x-3y, to be the numbers; and.. the numbers are ± 14, ± 10, ± 6, ± 2. 6. The sum of the first and second of four numbers in geometrical progression is 15, and the sum of the third and fourth is 60. Required the numbers. Let x, xy, xy', xy', be the numbers; 7. The sum of four numbers in geometrical progression is equal to the common ratio + 1; and the first term = 17 17 17 8. A regiment of militia was just sufficient to form an equilateral wedge. It was afterwards doubled by the supplementary, but was still found to want 385 men to complete a square containing 5 more men in a side, than in a side of the wedge. How many did the regiment at first contain? Let the number of men in a side of the wedge; .. (Alg. 212.) (x + 1). 2 = the number of men in the wedge; 2 or x2 + x + 385 = x2 + 10x + 25; .. by transposition, 360 = 9x, and 40 = x; .. the number of men = 820. |