14. Required the sum of n times of the series

13, 23, 3, 4, 5, 6”, &c. ; 1, 8, 27, 64, 125, 216, &c.

Here 1, 8, 27, 64, 125, 216, &c., given series.

7n(n-1), 12n(n−1)(n−2), 6n(n−1)(n—2)(n—3)

=n+

+

2

7n2-7n

n1—6n3+11n2-6n

=n+

·+2n3—6n2+4n+ 2

4

4n 14n2-14n, 8n3-24n+16n, n1-6n3+11n3-6n

4

+

+

+

4

4

4

-= sum of n terms, as required.

4

n1+2n3+n2 n2(n+1)2

4

15. What is the number of cannon-shot in a square pile, the bottom row consisting of 25 shot*? Ans. 5525.

16. I have 10 square house-lots, whose sides measure 5, 6, 7, 8, 9, &c., rods, respectively. What is their value, at 25 cents per square foot? Ans. $24,162.183.

* Shots and shells are generally piled in three different forms, called triangular, square, or oblong piles, according as their base is either a triangle, a square, or a rectangle.

A square pile is formed by the continual laying of square, horizontal courses of shot, one above another, in such a manner as that the sides of their courses decrease by unity from the bottom to the top row, which ends also in one shot.

17. There are 5 cubical blocks of marble, whose sides measure, respectively, 2, 3, 4, 5, and 6 feet? What is their value at $2.75 per cubic foot?

Ans. $1210.

18. What is the number of shot in a square pyramidical pile, whose side at the base contains 100 shot?

19. What is the sum of 20 terms of the 53, 63, &c.?

Ans. 338350.

series 13, 23, 33, 43,

Ans. 44100.

20. What is the sum of 20 terms of the series 14, 24, 34, 44, 54, 64, &c.? Ans. 722666.

PROBLEM V.

342. To find a fraction that will express the value of a geometrical series to infinity.

In Art. 284 we find that the sum of an infinite series is obtained by the following formula:

and, by this formula, we may find the sum of algebraic series.

EXAMPLES.

1. What is the sum of the series 1+a+a2+a3+a*, &c., carried to infinity?

1

Ans.

By the above formula, the first term of the series will be the numerator of the fraction, and the denominator is obtained by subtracting the second term from the first.

2. What fraction will express the exact value of the series 1+5+25+125, &c., to infinity?

1

Ans.

1-5°

3. What fraction will express the infinite series 1-a+a2—a3 +a+-a, &c.?

1

Ans.

h bh b2h

a

a2

a3

4. What fraction will express the series + + +, &c., to

infinity?

1.1 1 1

5. What is the sum of the series

+ +=+=+, &c., to х x2 x3 x4

1

infinity?

6. What fraction will express the series 1+2+4+8+16, &c.,

Ans.

8. What fraction will express the value of 1+1+1+1, &c.,

1

to infinity?

Ans.

x2 x3

9. Express by a fraction the value of the series x++

a

1

1 1

11. Required the sum of the series 12+2.3+3.4

2.3 +34 +, &c.,

Ans. 1.

1 1 1 + +: + +, 1.4 2.5 3.6 4.7

1, 1 1
+

1.32.4 3.5+, &c.,

2

3.5

3

Ans.

4'

+,

1

Ans.

12

SECTION XXXVIII.

CUBIC EQUATIONS, CONTAINING ONLY THE THIRD AND SECOND POWERS.

ART. 343. Any numerical equation, containing only the third and second powers of the unknown quantity, and having one rational root, may be reduced by rendering both of its members perfect squares, and extracting the square root of both sides; completing the operation by former rules. The only difficulty lies in multiplying the equation by such a number that, after adding to each side the fourth power of the unknown quantity, and the second power with a coefficient easily determined, both sides will be perfect squares. This multiplier must be ascertained by trial; for, though a general formula might be given for obtaining it, yet it would be so complicated as to be of no practical use. It may be either an integer or a fraction, and is positive or negative according to the sign of the known quantity.

Though there always is such a multiplier whenever the unknown quantity has one rational value, yet, when the numbers are very large, or the equation is very complicated, it may not be readily found, and the process of trial may become too tedious to be of service. Whenever the equation does not contain too large numbers, the pupil will find little difficulty, if he thoroughly understands the following

RULE. Divide both sides of the equation by the coefficient of the unknown cube, if it have any expressed. Place the third power of the unknown quantity on one side of the equation, and the second power, with the known quantity, on the other. Multiply both sides by the number nearest to unity which will make the known quantity a positive square; or, which is the same thing, separate the known number into two factors, one of which shall be the greatest square contained in it, and multiply both sides by the other factor.

Multiply the last equation by 4; add the fourth power of the

unknown quantity, and the second power, with a coefficient equal to the square of half the coefficient of the third power, to each side; and extract the square root of both sides, if possible. By taking like signs of the two members of the equation in evolving, we shall obtain one root; and, by taking unlike signs, the other two may be found by quadratic equations.

But, if that member of the equation which contains the known quantity is not a perfect square, substitute 1, 9, 16, 2, §, 1, §, or some other square number, in the place of 4, and proceed as above, till, by trial, a number is found which will accomplish the object.

NOTE.-1. The sum of the three values of the unknown quantity should always be equal to the coefficient of the second power in the original equation, after dividing by the coefficient of the cube, and placing it on the same side with the known quantity, opposite the positive cube; hence, if two values were known, the other might easily be found.

2. When one of the values is known, the others might be found by the usual method; bringing all the terms of the original equation to the same side, and dividing by the difference between the unknown quantity and its known value, reducing, by quadratic equations, the equation thus produced. But the three values are here given directly, by using the different signs in evolving, thus rendering the solution shorter, and more satisfactory. It is evident that, in extracting the square root of an equation, both sides may be considered positive, or both negative, or either one positive and the other negative. Thus the square root of the equation 4a2—8ab4b2—a2+2cd+d2, is +(2a—2b)=+(c+d), or —(2a—2b)

−(c+d), or +(2a—2b)=—(c+d), or —(2a—2b)=+(c+d). But, if both sides take like signs, the result will be the same, whether they are both positive or both negative, as the signs of both sides of an equation may always be changed; while, if they take unlike signs, a different equation will be produced, it making no difference which side is positive. Hence, there are but two results that can be obtained, and we have preferred to express them, in the following examples, by the same method as in quadratic equations; prefixing the sign to the right-hand member of the equation produced by evolution.

3. By observing whether the root of the known quantity is greater or less than half the coefficient of the second power on the same side, if we also notice the sign, we may usually know whether the multiplier we have used is too small or too large. When there are two rational values of the unknown quantity, of course the third will be rational, and there will be three different multipliers, which will answer our purpose, thus giving three different solutions for the same example.