Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

RULE,

For finding the last term.

Multiply the common difference by one less than the number of terms, and add the first term to the product, if the series be increasing; but if the series be decreasing, subtract the product from the first

term.

Note.--All questions may be considered to be in increasing pregression, unless mentioned otherwise in the question.

2. What is the last term of a series, which has 1001 terms, a common difference of 7, and the first term 19 ? Ans. 7019. 3. Given the first term of a series 111'111, the number of terms 1'111, and the common difference 11; what is the last term, if the series decrease? Ans. 98901.

4. If a vessel sail 100 days successively, and sail 100 miles the first day, how far will she sail the last day, allowing she decreases each day's sail of a mile? Ans. 67 miles.

[ocr errors]
[blocks in formation]

To find the sum of the terms.

10. A man paid a debt at 41 several payments. The first payment was $26,01, and each succeeding payment increased in arithmetical progression with a common difference of $1,85; what was the sum of all the payments, that is, what was the debt?

Here 185 cents X 40=7400 cents; and 7400 cents+2601 cents=10001 cents the last payment. Then 10001 cents+2601=12602 cents the sum of the first and last payments, and 12602 × 41, the number of payments=516682 cents double the sum of all the payments. Ans. $2583,41.

Illustration.-The sum of any two terms of an arithmetical series equally distant from the centre, is equal to double the sum, which all the terms will average; or if the number of terms be odd, equal to double the middle term, consequently multiplying the sum of the extremes by the number of terms, gives double the sum of the series.

RULE.

Multiply the sum of the extremes by the number of terms, and divide the product by 2; or multiply half the sum of the extremes by the number of terms.

11. If the first term of an arithmetical series be 2, the last term 122, and the number of terms 29, what is the sum of all the terms?

Ans. 1798.

12. What is the sum of 1000 terms in arithmetical progression,-the last term being 500, and the common difference, if the series decrease?

Ans. 583250.

13. Required the sum of 990 terms of the numbers 2, 5, 8, 11, &c. ? Ans. 1470645.

Given the following examples to find the sum of the

series in each.

[blocks in formation]

Required the sum of the series in the following ex

amples.

First terms. Com. diff. No. terms. Ans.

[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

Find the sum of all the terms in the following examples, in decreasing arithmetical progression. Com. diff. No. terms. Ans.

ཉན

First terms.

[blocks in formation]

6,

[blocks in formation]
[blocks in formation]

PROBLEM III.

To find the common difference.

RULE.

Divide the difference of the extremes by one less than the number of terms.

Illustration.--The greater term is made up of the less, and the common difference taken as many times as the series contains terms wanting one; therefore by subtracting the less from the greater extreme, and dividing the difference by one less than the number of terms, the quotient will be the common difference of the terms.

24. The extremes of an arithmetical series are 3 and 97, and the number of terms 48; what is the common difference? Ans. 2.

25. If a man perform a journey in 8 days, increasing each day's travel by a certain number of miles, and travel the first day 21 miles, and the last 70;— how many miles does he increase daily?

PROBLEM IV.

To find the number of terms.

RULE.

Ans. 7.

Divide the difference of the extremes by the common difference, and add one to the quotient.

Illustration.-The greatest term is made up of the common difference added together as many times

as the series has terms wanting one, together with the less term; therefore by subtracting the less term from the greater, and dividing the remainder by the common difference, the quotient expresses the number of times the common difference was added together to make the dividend, which is the difference between the extremes.

25. The extremes of an arithmetical series are 8 and 88, the common difference 10; what is the number of terms? Ans. 9.

26. If a boy buy a number of oranges, and give 17 cents for the first, 15 cents for the second, &c. decreasing the price of each, 1 cents until the last cost but 2 cents;-how many oranges did he buy? Ans. 11.

27. A spendthrift spent his estate, amounting to $2873,10, in 15 years, spending each year $20,22 more than in the preceding; how much did he expend the 9th year? Ans. $211,76. Questions to be performed by the principles contained in the preceding problems.

28. If two men start from the same place at the same time, one travelling 21 miles each day, the other 3 miles the first day, 5 the second, increasing his speed cach day by the excess of two miles; how many days before the latter will come up with the former, and how many miles will each travel? 19 days. Ans. 399 miles.

{

29. Estimating the distance between London and Boston at 3000 miles, if a ship perform a voyage from one city to the other in 26 days,-sailing the first day 80 miles, and increase each day's sail in arithmetical progression; how far will she sail the last day; what will be the daily increase; and how much farther will she sail the 25th day of the voyage than the second?

Ans.

[blocks in formation]

25th day's sail more than the 2d, miles.

65

30. Two divisions of labourers commence work upon a canal at the same time, 500 rods apart; one division completes one half of the intervening distance in 7 days; the other the remaining half in 9 days. Now if each division increase each day's work 4 rods more than the last, how many rods does each division complete in each day?

Ans.

The first, 23, 274, 314, 35, 39, 43, and 47.

The second, 117, 157, 197 233, 277, 317, 357, 397, and 437.

The methods which have been explained for finding the last term, sum of the terms, common difference and number of terms, may be more clearly illustrated by the assistance of the adjoining plate.

[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
« ΠροηγούμενηΣυνέχεια »