6. What is the depth of a cubical cistern that holds 200 barrels of water?

7. Find the length of a cubical vessel that will hold 4000 gallons of water.

ROOTS OF HIGHER DEGREE.

822. Any root whose index contains no other factors than 2 or 3 may be extracted by means of the square and cube roots.

If any power of a given number is raised to any required power, the result is that power of the given number denoted by the product of the two exponents. (801.) Conversely, if two or more roots of a given number are extracted, successively, the result is that root of the given number denoted by the product of the indices. 1. What is the 6th root of 2176782336 ?

OPERATION.

2176782336 = 46656

46656

= 36

Or,

2176782336 = 1296 1296

= 36

ANALYSIS.-The index of the required root is 6'= 2 × 3; hence extract the square root of the given number, and the cube root of this result, which gives 36 as the 6th or required root. Or, first find the cube root of the given number, and then the square root of the result.

RULE.-Separate the index of the required root into its prime factors, and extract successively the roots indicated by the several factors obtained; the final result will be the required root.

2. What is the 4th root of 5636405776 ?

3. What is the 8th root of 1099511627776?

4. What is the 6th root of 25632972850442049 ?

5. What is the 9th root of 1.577635?

For further practical applications of Involution and Evolution, see "Mensuration."

PROGRESSIONS

823. An Arithmetical Progression is a succession of numbers, each of which is greater or less than the preceding one by a constant difference.

Thus, 5, 7, 9, 11, 13, 15, is an arithmetical progression.

824. The Terms of an arithmetical progression are the numbers of which it consists. The first and last terms are called the Extremes, and the other terms the Means.

825. The Common Difference is the difference between any two consecutive terms of the progression.

826. An Increasing Arithmetical Progression is one in which each term is greater than the preceding one.

Thus, 1, 3, 5, 7, 9, 11, is an increasing progression.

827. A Decreasing Arithmetical Progression is one in which each term is less than the preceding one.

Thus, 15, 13, 11, 9, 7, 5, 3, 1, is a decreasing progression.

828. The following are the quantities considered in arithmetical progression :

1. The first term (a).

3. The common difference (d).

2. The last term (7).

4. The number of terms (n).

5. The sum of all the terms (s).

WRITTEN EXERCISES.

829. To find one of the extremes, when the other extreme, the common difference, and the number of terms are given.

1. The first term of an increasing progression is 8, the common difference 5, and the number of terms 20; what is the last term?

OPERATION.

20 1

19

19 x 58103 = 7.

ANALYSIS.-The 2d term is 8+5; the 3d term is 8+ (5 × 2) the 4th term is 8+(5 × 3); and so on. Hence 8+ (19 × 5) or 103 is the 20th or last term. 2. The last term of an increasing progression is 103, the common difference 5, and the number of terms 20; what is the first term?

OPERATION.

20-1= 19

103 19 x 58 = a

ANALYSIS.-The 1st term must be a number to which, if 19 × 5 be added, the sum shall be 103; hence, if 19 × 5 is subtracted from 103, the remainder is the first term.

3. The first term of a decreasing progression is 203, the common difference 5, and the number of terms 40; what is the last term?

4. The last term of a decreasing progression is 1, the common difference 2, and the number of terms 9 ; what is the first term?

RULE.-I. If the given extreme is the less, add to it the product of the common difference by the number of terms less one.

II. If the given extreme is the greater, subtract from it the product of the common difference by the number of terms less one.

5. The first term of an increasing progression is 5, the common difference 4, and the number of terms 8; what is the last term?

6. The first term of an increasing progression is 2, and the common difference 3; what is the 50th term ?

7. The first term of a decreasing progression is 100, and the common difference 7; what is the 13th term? 8. The first term of an increasing progression is, the common difference, and the number of terms 20; what is the last term?

830. To find the common difference, when the extremes and number of terms are given.

1. The extremes of a progression are 8 and 103, and the number of terms 20; what is the common difference?

103

OPERATION.

8195 = d

ANALYSIS.-The difference between the extremes is equal to the product of the common difference by the number of terms less one (829); hence the common difference is 15, or 5.

2. The extremes of a progression are 1 and 17, and the number of terms 9; what is the common difference?

RULE.-Divide the difference between the extremes by the number of terms less one.

3. The extremes are 3 and 15, and the number of terms 7; what is the common difference?

4. The extremes are 1 and 51, and the number of terms 76; what is the common difference?

5. The youngest of ten children is 8, and the eldest 44 years old; their ages are in arithmetical progression. What is the common difference of their ages ?

6. The amount of $800 for 60 years, at simple interest, is $4160. What is the rate per cent. ?

7. The extremes are 0 and 24, and the number of terms 18; what is the common difference?

831. To find the number of terms, when the extremes and common difference are given.

1. The extremes of a progression are 8 and 103, and the common difference 5; what is the number of terms?

therefore 19+1 or 20 is the number of terms.

2. The extremes of a progression are 1 and 17, and the common difference 2; what is the number of terms?

RULE.-Divide the difference between the extremes by the common difference, and add one to the quotient.

3. The extremes are 5 and 75, and the common difference is 5; what is the number of terms?

4. The extremes are and 20, and the common difference is 6; what is the number of terms?

5. A laborer received 50 cents the first day, 54 cents the second, 58 cents the third, and so on, until his wages were $1.54 a day; how many days did he work?

6. In what time will $500, at 7 per cent. simple interest, amount to $885?