1. Find the sum of 6 terms of the series whose first term

Since the two series are the same, their sum is twice the first series. But their sum is obviously as many times 12, (the sum of the extremes), as there are terms. Hence, the

Rule for Case III.—Multiply the sum of the extremes by the number of terms; half the product will be the sum of the series.

2. The extremes are 2 and 50; the number of terms, 24: find the sum of the series. Ans. 624.

3. How many strokes does the hammer of a clock strike in 12 hours?

Ans. 78.

4. Find the sum of the first ten thousand numbers in the series, 1, 2, 3, 4, 5, &c. Ans. 50005000.

5. Place 100 apples in a right line, 3 yd. from each other, the first, 3 yd. from a basket: what distance will a boy travel who gathers them singly and places them in the basket? Ans. 17 mi. 380 yd. 6. A traveled one day 30 mi., and each succeeding day a quarter of a mile less than on the preceding day: how far did he travel in 30 days? Ans. 7911 mi.

7. A body falling by its own weight, if not resisted by the air, would descend in the 1st second a space of 16 ft. Í in.; the next second, 3 times that space; the 3d, 5 times that space; the 4th, 7 times, &c.: at that rate, through what space would it fall in 1 min. ? Ans. 57900 ft.

XXVI. GEOMETRICAL PROGRESSION. ART. 305. A Geometrical Progression, or Series, is a series of numbers increasing by a common multiplier, or decreasing by a common divisor. Thus,

1,

31

9, 27,

48, 24, 12, 6,

81, is an increasing geometric series. 3, is a decreasing geometric series.

REVIEW.-303. How find the number of terms, when the extremes and common difference are given ?

The common multiplier or com. divisor, is called the ratio. In the 1st of the above series, the ratio is 3; in the 2d, 2.

The numbers forming the series are the terms; the first and last terms are extremes; the others, means.

ART. 306. In every geometric series, 5 things are considered: 1st, the first term; 2d, the last term; 3d, the number of terms; 4th, the ratio; 5th, the sum of all the terms.

CASE I.

ART. 307. To find the LAST TERM, when the first term, the ratio, and the number of terms are given.

1. The first term of an increasing geometric series is 2; the ratio 3; what is the 5th term?

SOLUTION.-The first term is 2; the second, 2X3; the third, 2X3X3; the fourth, 2×3×3×3; and

The fifth, 2×3×3×3×3=2×31 =2×81 =162. Ans.

Observe that each term after the first, consists of the first term multiplied by the ratio taken as a factor as many times less one, as is denoted by the number of the term. Thus, the fifth term consists of 2 multiplied by 3 taken four times as a factor. But 3, taken 4 times as a factor, is (Art. 277) the 4th power of 3. Hence, The fifth term is equal to 2, multiplied by the 4th power of 3.

2. The first term of a decreasing geometric series is 192; the ratio 2; what is the fourth term?

SOLUTION.-The 2d term is 192÷2; the 3d is 192 divided by 2 X 2; the 4th is 192 divided by 2 × 2 × 2; that is, 192231928-24, Ans. The required term is found by dividing the first term by the ratio raised to a power whose exponent is 1 less than the number of the term. Hence, the

Rule for Case I.-Raise the ratio to a power (Art. 279) whose exponent is one less than the number of terms.

If the series be increasing, MULTIPLY the 1st term by this power, and the product will be the last term; if decreasing, DIVIDE the 1st term by the power, and the quotient will be the last term.

REVIEW.-304. What is Case 3? What is the Rule for Case 3? 805. What is a geometrical series? Give examples. What the ratio? What the extremes? The means? 306. What five things are considered?

NOTE. In finding high powers of the ratio, the operation may often be shortened by observing that the product of any two powers of a number, will give that power of the number which is denoted by the sum of their exponents. Thus,

The third power multiplied by the fourth power, will produce the seventh power.

23X248X16=128=27.

3. The first term of an increasing series is 2; the ratio, 2; the number of terms, 13: find the last term. Ans. 8192.

4. The first term of a decreasing series is 262144; the ratio, 4; number of terms, 9: find the last term. Ans. 4. 5. The first term of an increasing series is 10; the ratio, 3: what the tenth term? Ans. 196830.

6. What the 35th term of an increasing series, whose first term is 1, and ratio, 2? Ans. 17179869184. 7. Find the 35th term of an increasing series, the 1st term, 1; ratio, 3. Ans. 16677181699666569.

CASE II.

ART. 308. To find the sum of all the terms of a geometric

series.

1. To obtain a General Rule, let us find the sum of 5 terms of the geometric series, whose 1st term is 4, and ratio 3.

SOLUTION. Write the terms of the series as below; then multiply each term by the ratio, and remove the product one term toward the right: thus,

4+12+36 +108+ 324

sum of the series. = sum X 3.

12+36 +108 +324 +972 = Since the upper line is once the sum of the series, and the lower three times the sum, their difference is twice the sum; hence,

If the upper line be subtracted from the lower, and the remainder divided by 2, the quotient will be the sum of the series.

Performing this operation, we have 972-4-968; which, divided by 2, the quotient is 484, the required sum.

In this process, 972 is the product of the greatest term of the given series by the ratio; 4 is the least term, and the divisor 2, is equal to the ratio less one.

REVIEW.-307. What is Case 1? What the Rule? NOTE. How are high powers of the ratio most easily found?

Rule for Case II.—Multiply the greatest term by the ratio; from the product subtract the least term, and divide the remainder by the ratio less 1; the quotient will be the sum of the series.

NOTE.—When a series is decreasing, and the number of terms infinite, the last term is naught. In finding the sum by the rule, observe that the ratio is greater than 1.

2. The first term is 10; the ratio, 3; the number of terms, 7: what is the sum of the series? Ans. 10930.

3. A gave to his daughter on New Year's day $1; he doubled it the first day of every month for a year: what sum Ans. $4095.

did she receive?

4. I sold 1 lb. of gold at 1 ct. for the 1st oz., 4 for the 2d, 16 for the 3d, &c.: what the sum? Ans. $55924.05

5. A sold a house having 40 doors, at 10 cts. for the 1st door, 20 for the 2d, 40 for the 3d, and so on: how much did he receive? Ans. $109951162777.50 for the 1st nail in his shoes, what was the price, there being Ans. $9265100944259.20

6. B bought a horse at 1 ct. 3 for the 2d, 9 for the 3d, &c.: 32 nails?

7. Find the sum of an infinite series, the greatest term .3; the ratio, 10; that is, of +80+000, &c.

3

Ans..

8. Find the sum of an infinite series, greatest term 100; ratio 1.04

9. The sum of the infinite series 1, 1, 27, 10. The sum of the infinite series 1, 1, 1,

Ans. 2600.

&c.

Ans.

&c.

Ans. 1.

XXVII. PERMUTATION.

ART. 309. Permutation teaches the method of finding in how many different positions any given number of things may be placed.

Thus, the two letters a and b can be placed in two positions, ab and ba; but if we add a third letter c, three positions can be made with each of the two preceding: thus,

Cab, acb, abc, and cba, bca, bac, making 2×3=6 positions.

By taking a fourth letter d, four positions can be made out of each of the six positions, making 6×4-24 in all.

Rule for Permutation.- Multiply together the

num

bers, 1, 2, 3, &c., from 1 to the given number; the last product will be the required result.

1. In how many different ways may the digits 1, 2, 3, 4, and 5 be placed? Ans. 120.

2. What number of changes may be rung on 12 bells? Ans. 479001600. 3. What time will 8 persons require to seat themselves differently every day at dinner, allowing 365 days to the year? Ans. 110yr. 170 da.

4. Of how many variations do the 26 letters of the alphabet admit? Ans. 403291461126605635584000000.

XXVIII. MENSURATION.

TO TEACHERS.-As this short article on Mensuration is intended for pupils who may not have an opportunity of studying a more extensive course, only the more useful parts are presented.

The definitions and illustrations are given in plain and familiar terms, not with a view to mathematical precision.

ART. 310. DEFINITIONS,

1. An ANGLE is the inclination of two straight lines meeting in a point, which is called the VERTEX. It is the degree of the opening of the lines.

2. When one straight line stands on another so that it makes with it two equal angles, each of these angles is a RIGHT ANGLE; and the straight line which stands on the other is said to be PERPENDICULAR to it, or at RIGHT ANGLES to it.

angle.

Acute 3. An OBTUSE ANGLE is greater than a right angle: and an ACUTE ANGLE is less than a right angle.

D

NOTE. An angle is named by 3 letters, the middle one being placed at the vertex, and the other two on the lines which form

REVIEW.-308. What is Case 2? What the Rule? NOTE. What is the last term of a decreasing series of which the number of terms is infinite? 309. What is Permutation? What the Rule?