## Algebraical Problems, Producing Simple and Quadratic Equations, with Their Solutions; Designed as an Introduction to the Higher Branches of Analytics: to which is Added an Appendix, Containing a Collection of Problems on the Nature and Solution of Equations of Higher Dimensions |

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**Harmonic Progression**. 1. EXPLAIN the nature of**harmonic progression**; and continue in both directions the series , 2 , 3 , 6 . 2. Continue the**harmonic progression**.... 3 , 4 , 6 .... up- wards and downwards . How far can it be ... Page 416

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**progression**be known ; the intervening series may be found . 10. Insert two**harmonic**means between 2 and 4 ; two between 6 and 24 ; four between 2 and 12 ; six between 1 and 20 ; and n between x and y . a 11. Insert n**harmonic**means ... Page 417

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**harmonic progression**; prove that the first has to the second the same ratio which the third has to the fourth . 18. The sum of three terms of an**harmonic progression**, whose first term is , is = ; determine the progression and 1 ... Page 427

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**harmonic progression**, and p , q , r , integer numbers , then r is the square of the greatest root . Apply this to solve the equation a3 — 23x2 + 135x - - - 81. If the roots of the equation - px2 + qxr = o , be in**harmonic progression**... Page 428

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**harmonic progression**, then will the greatest and least be and - n ✓ ( n + 1 ) . L √ ( n + 1 ) . P − √ { 3 . ( n − 1 ) 2 . P2 — 6n . ( n − 1 ) . QL } ' n . √ ( n + 1 ) . L √ ( n + 1 ) .P + √ { 3 . ( n − 1 ) 2 . P ' — 6n . ( n ...### Other editions - View all

### Common terms and phrases

a²x² addition answer the conditions arithmetic series arithmetical progression casks common difference completing the square containing cost cube digits distance equal equation a³ equation of fractions extracting the root extracting the square find the values gain geometric series geometrical progression Given 3x Given x² greater guineas harmonic progression least common multiple length less Let 2x Let the number number of days number of gallons number of miles number of shillings number of terms number of yards pence pieces problem proportion px² Quadratics received Required the number second equation sheep shil sold square root squaring both sides Substituting this value subtraction third three numbers Transform the equation transposition travelled unknown quantity values of x wheat whence whole number x²y xy²

### Popular passages

Page 24 - In one of the given equations obtain the value of one of the unknown quantities in terms of the other unknown quantity; Substitute this value in the other equation and solve.

Page 251 - The fore wheel of a carriage makes 6 revolutions more than the hind wheel in going 120 yards ; but if the periphery of each wheel be increased one yard, it will make only 4 revolutions more than the hind wheel in the same space.

Page 378 - From two places at a distance of 320 miles, two persons, A and B, set out at the same time to meet each other. A travelled 8 miles a day more than B, and the number of days in which they met was equal to half the number of miles B went in a day. How many miles did each travel, and how far per day ? 20.

Page 2 - Any quantity may be transposed from one side of an equation to the other, if, at the same time, its sign, be changed.

Page 371 - A detachment of soldiers from a regiment being ordered to march on a particular service, each company furnished four times as many men as there were companies in the...

Page 231 - There are two square buildings, that are paved with stones, a foot square each. The side of one building exceeds that of the other by 12 feet, and both their pavements taken together contain 2120 stones. What are the lengths of them separately 1 Ans.

Page 157 - His head weighed as much as his tail and half his body, and his body weighed as much as his head and tail together. What was the weight of the fish ? Let 2x = the weight of the body in pounds.

Page 375 - A gentleman bought two pieces of silk, which, together, measured 36 yards. Each of them cost as many shillings per yard as there were yards in the piece, and their whole prices were as 4 to 1. What were the lengths of the pieces ? Solution.

Page 365 - There is a cistern, into which water is admitted by three cocks, two of which are of exactly the same dimensions. When they are all open, five-twelfths of the cistern is filled in...

Page 188 - Prob. 3. Find two numbers, the greater of which shall be to the less, as their sum to 42 ; and as their difference to 6.