GEOMETRICAL PROGRESSION.

§ 193. If we take any number, as 3, and multiply it continually by any other number, as 2, we form a series of numbers, thus,

3 6 12 24 48 96 192, &c., in which each number is formed by multiplying the number before it, by 2.

This series may also be formed by dividing continually the largest number 192 by 2. Thus,

192 96 48 24 12 6 3.

A series formed in either way is called a Geometrical Series, or a Geometrical Progression, and the number by which we continually multiply or divide, is called the

common ratio.

When the series is formed by multiplying continually by the common ratio, it is called an ascending series; and when it is formed by dividing continually by the common ratio, it is called a descending series.

Thus,

is an ascending series.

is a descending series.

3 6 12 24 48 96 192 192 96 48 24 12 6 2 The several numbers are called terms of the progression. The first and last terms are called the extremes, and the intermediate terms are called the means.

Q. How do you form a Geometrical Progression? What is the common ratio? What is an ascending series? What is a descending series? What are the several numbers called? What are the first and last terms called? What are the intermediate terms called?

§ 194. In every Geometrical, as well as in every Arithmetical Progression, there are five things which are considered, any three of which being given or known, the remaining two can be determined.

By considering the manner in which the ascending progression is formed, we see that the second term is obtained by multiplying the first term by the coinmon ratio; the 3rd term by multiplying this product by the common ratio, and so on, the number of multiplications being one less than the number of terms. Thus,

3=1 1st term, 3x2=6 2nd term, 3x2x2=12 3rd term,

3×2×2×2=24 4th term, &c. for the other terms. But 2×2=22, 2×2×2=23, and 2×2×2×2=24.

Therefore, any term of the progression is equal to the first term multiplied by the ratio raised to a power 1 less than the number of the term.

Q. In every Geometrical Progression, how many things are considered? What are they?

CASE I.

Having given the first term, the common ratio, and the number of terms, to find the last term.

RULE.

Raise the ratio to a power whose exponent is one less than the number of terms, and then multiply the power by the first term, the product will be the last term.

EXAMPLES.

1. The first term is 3 and the ratio 2; what is the 6th term?

2×2×2×2×2=25=32

3 1st term

Ans. 96

2. A man purchased 12 pears: he was to pay 1 farthing for the first, 2 farthings for the 2nd, 4 for the 3rd, and so on doubling each time: what did he pay for the last? Ans. £2 2s. 8d. 3. A gentleman dying left nine sons, and bequeathed his estate in the following manner: to his executors £50;

his youngest son to have twice as much as the executors, and each son to have double the amount of the son next younger: what was the eldest son's portion?

Ans. £25600. 4. A man bought 12 yards of cloth, giving 3 cents for the 1st yard, 6 for the 2nd, 12 for the 3rd, &c.: what did he pay for the last yard? Ans. $61,44.

CASE II.

§ 195. Having given the ratio and the two extremes to find the sum of the series.

RULE.

Subtract the less extreme from the greater, divide the remainder by 1 less than the ratio, and to the quotient add the greater extreme: the sum will be the sum of the series. Q. How do you find the sum of the series?

EXAMPLES.

1. The first term is 3, the ratio 2, and last term 192: what is the sum of the series?

192-3-189 difference of the extremes,

2-1-1)189(189; then 189+ 192=381 Ans.

2. A gentleman married his daughter on New Year's day, and gave her husband 1s. towards her portion, and was to double it on the first day of every month during the year what was her portion?

Ans. £204 15s.

3. A man bought 10 bushels of wheat on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2nd, 9 for the 3rd, and so on to the last: what did he pay for the last bushel and for the 10 bushels ?

Ans. last bushel $196,83, total cost $295,24. 4. A man has 6 children; to the 1st he gives $150, to the 2nd $300, to the 3rd $600, and so on, to each twice as much as the last: how much did the eldest receive and what was the amount received by them all?

Ans. Eldest $4800, total $9450

APPENDIX.

MENSURATION.

§ 196. A triangle is a figure bounded by three straight lines. Thus, BAC, is a triangle.

The three lines BA, AC, BC, are called sides: and the three corners, B, A, and C, are called angles. The side BC is called the base.

B

When a line like AD is drawn making the angle ADB equal to the angle ADC, then AD is said to be perpendicular to BC, and AD is called the altitude of the triangle. Each triangle BAD or DAC is called a right angled triangle. The side BA or the side AC, opposite the right angle, is called the hypothenuse.

The area or content of a triangle is equal to half the product of its base by its altitude.

EXAMPLES.

1. The base of a triangle is 40 yards and the perpen dicular 20 yards: what is the area?

We first multiply the base by the altitude and the product is square yards, which we divide by 2 for the

area.

OPERATION.

40

20

2)800

Ans. 400 square yards.

2. In a triangular field the base is 40 chains and the perpendicular 15 chains: how much does it contain? (see § 64.) Ans. 30 acres 3. There is a triangular field of which the base is 35 rods and the perpendicular 26 rods: what is its content? Ans. 2A. 3R. 15P.

4. What is the area of a square field of which the sides are each 33,08 chains?

Ans. 109A. 1R. 28P+.

5. What is the area of a square piece of land of which the sides are 27 chains?

6. What is the area of a square the sides are 25 rods each?

Ans.

piece of land of which

Ans. 3A. 3R. 25P.

§ 197. A rectangle is a four-sided figure like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal.

The area or content of a rectangle is equal to the length multiplied by the breadth.

EXAMPLES.

1. What is the content of a rectangular field the length of which is 40 rods and the breath 20 rods?

Ans. 5 acres. 2. What is the content of a field 40 rods square?

Ans. 10 acres.

Ans.

3. What is the content of a rectangular field 15 chains long and 5 chains broad. 4. What is the content of a field 25 chains long by 20 chains broad? 5. What is the content of a field 27 chains long and 9 rods broad. Ans. 6A. OR. 12P.

§ 198. A circle is a portion of a plane bounded by a curved line, every part of which is equally distant from a certain point within, called the centre.

The curved line AEBD is called the circumference: the point C the centre; the line AB passing through the centre, a diameter, and CB the radius.

A

Ans. 50 acres.

E

B

The circumference AEBD is 3,1416 times greater than the diameter AB. Hence, if the diameter is 1, the circumference will be 3,1416. Hence, also, if the diameter is known, the circumference is found by multiplying 3,1416 by the diameter.