Rule.-Divide the difference of the two extremes by the common difference, and add 1 to the quotient: the sum will be the number of terms.

Examples.

1. A farmer sold a number of bushels of wheat; it was agreed that, for the first bushel, he should receive 50 cents, and an increase of 9 cents for each succeeding bushel, and for the last, he received $500: how many bushels did he sell?

2. A person proposes to make a journey, and to travel 15 miles the first day, and 33 miles the last, with a daily increase of 1 miles in how many days did he make the journey, and what was the whole distance traveled?

3. I owe a debt of $2325, and wish to pay it in equal installments, the first payment to be $575, the second, $500, and decreasing by a common difference, until the last payment, which is $200 what will be the number of installments?

GEOMETRICAL PROGRESSION.

399. A GEOMETRICAL PROGRESSION is a series of terms, each of which is derived from the preceding one, by multiplying it by a constant number. The constant multiplier, is called the ratio of the progression.

400. AN INCREASING SERIES is one whose ratio is greater than 1:

A DECREASING SERIES is one whose ratio is less than 1. Thus,

1, 2, 4, 8, 16, 32, &c.-ratio 2-increasing series: 32, 16, 8, 4, 2, 1, &c.-ratio decreasing series.

The several numbers resulting from the multiplication, are called terms of the progression. The first and last terms are

called the extremes, and the intermediate terms are called

means.

401. In every Geometrical, as well as in every Arithmetical Progression, there are five parts:

1st, The first term;

2d, The last term;
3d, The common ratio;

4th, The number of terms;

5th, The sum of all the terms;

If any three of these parts are known, or given, the remaining ones can be determined.

CASE I.

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

1. The first term is 4, and the common ratio 3: what is the 5th term?

multiplications being 1 less than the number of terms: thus,

3 × 3 × 3 × 3 × 4 = 324, 5th term.

Therefore, the last term is equal to the first term multiplied by the ratio raised to a power whose exponent is 1 less than the number of terms.

Rule.-Raise the ratio to a power whose exponent is 1 less than the number of terms, and then multiply this power by the first term.

Examples.

1. The first term of a decreasing progression is 2187; the ratio is, and the number of terms 8: what is the last term?

2. The first term of an increasing geometrical series is 8, the ratio 5 what is the 9th term?

:

3. The first term of a decreasing geometrical series is 729, the ratio what is the 10th term?

4. If a farmer should sell 15 bushels of wheat, at 1 mill for the first bushel, 1 cent for the second, 1 dime for the third, and so on; what would he receive for the last bushel ?

5. A man dying left 5 sons, and bequeathed his estate in the following manner to his executors, $100; to his youngest son twice as much as to the executors, and to each son double the amount of the next younger brother: what was the eldest son's portion?

6. A merchant engaging in business, trebled his capital once in 4 years if he commenced with $2000, what would his capital amount to at the end of the 12th year?

7. A farmer wishing to buy 16 oxen of a drover, finally agreed to give him for the whole the cost of the last ox only. He was to pay 1 cent for the first, 2 cents for the second, and doubling on each one to the last how much would they cost him?

8. What is the amount of $500 for 3 years at 6 per cent. compound interest?

NOTE.-The ratio is 1.06.

CASE II.

403. Knowing the two extremes and the ratio, to find the sum of the terms.

1. What is the sum of the terms of the progression 2, 6, 18, 54, 162?

ANALYSIS.-If we multiply the terms of the progression by the ratio 3, we have a second progression, 6, 18, 54, 162, 486, which is 3 times as great as the first. If from this we subtract the first, the remainder, 486 — 2, will be 2 times as great as the first; and if this remainder be divided by 2, the quotient will be the sum of the terms of the first progression.

But 486 is the product of the last term of the given progression multiplied by the ratio; 2 is the first term; and the divisor 2, 1 less than the ratio: hence,

Rule.-Multiply the last term by the ratio; take the dif ference between this product and the first term, and divide the remainder by the difference between 1 and the ratio.

NOTE.-When the progression is increasing, the first term is subtracted from the product of the last term by the ratio, and the divisor is found by subtracting 1 from the ratio. When the progression is decreasing, the product of the last term by the ratio is subtracted from the first term, and the ratio is subtracted from 1.

Examples.

1. The first term of a progression is 4, the ratio 3, and the last term 78722: what is the sum of the terms?

2. The first term of a progression is 1024, the ratio, and the last term 4: what is the sum of the series?

3. What debt can be discharged in one year by monthly payments, the first being $2, the second $8, and so on to the end of the year; and what will be the last payment?

4. A gentleman being importuned to sell a fine horse, said

that he would sell him on the condition of receiving 1 cent for the first nail in his shoes, 2 cents for the second, and so on, doubling the price of every nail: the number of nails in each shoe being 8, how much would he receive for his horse?

5. A laborer agreed to thresh 64 days for a farmer, on the condition that he should give him 1 grain of wheat for the first day's labor, 2 grains for the second, and double each succeeding day what number of bushels would he receive, supposing a pint to contain 7680 grains; and what number of ships, each carrying 1000 tons burden, might be loaded, allowing 40 bushels to a ton?

:

ANALYSIS.

404. AN ANALYSIS is an examination of the separate parts of a proposition, and of the connection of those parts with each other.

In analyzing, we generally reason from a given number to its unit, and then from this unit to the required number.

The process is indicated by the relations which exist between the given and the required numbers, and pursued, step by step, independently of set rules.

1. If 12 yards of cloth cost $48.36, what will 7 yards cost? ANALYSIS.-One yard of cloth will cost as much as 12 yards: since 12 yards cost $48.36, one yard will cost of $48.36 = $4.03; 7 yards will cost 7 times as much as 1 or 7 times of $48.36 = $28.21; therefore, if 12 yards of cloth cost $48.36, 7 yards will cost $28.21.

yard,

or

2. If 27 pounds of butter will buy 45 pounds of sugar, how

much butter will 36 pounds of sugar buy?