had made, was equal to 55 more than A had paid altogether. How many payments had they made? Suppose we wish to find the sum of any number of terms of a series in arithmetical progression; for example, the sum of ten terms of the series, 4, 6, 8, 10, &c. Let 8 = the sum, Then 8=4+ 6 +8 + 10 + 12 + 14 + 16 +18+20 +22 And 822+20+18+16+14+12+10+8+6+4 2826+26+26+26+26+26+26+26+26+26 2810 X 26 10 X 26 2 Multiply the number of terms by 26, and divide by 2; but this number 26 is obtained, by adding together the first term 4 of the series, and the last term 22. We have, then, this rule for finding the sum of a series. RULE.-Add together the first and last terms of the series, multiply their sum by the number of terms, and divide by 2. 10. What is the sum of 20 terms of the series 1, 2, 3, 4, 5, &c. 11. How many strokes does the hammer of a clock strike in 12 hours? 12. What is the sum of 15 terms of a series, whose first term is 5, and common difference 2? 13. If a represent the first term of a series, r the common difference, and n the number of terms; what will represent the last term? and what will represent the sum of all the terms? 14. How long will it require a man to travel round the world, supposing the distance to be 24877 miles, if he should go 1 mile the first day, 2 the second, 3 the third, &c.? 15. If A sets out from C towards D, and travels 1 mile the first day, 2 the second, 3 the third, &c.; and, five days after, B sets out from C towards D, and travels every day 12 miles; how long before B will overtake A? SECTION LV. Geometrical Progression. If a series of terms increase or decrease, by a common multiplier or divisor, it is called a geometrical progression. 1, 2, 4, 8, 16, &c.; 20, 10, 5, 21⁄2, &c., are examples of series of this kind. The common multiplier or divisor, is called the ratio or quotient. 1. If a be the first, r the ratio, what will represent the second term? what the third? what the fourth? &c. 2. What formula will represent any series in geometrical progression, supposing a the first term, and r the ratio? Ans. a, ar, a r2, a r3, a ra, a r3, &c. 3. If a 1 and r = 2; what will the formula become? 4. If a = 4 and r = 3; what will the formula become? 5. If a=2 and r = ; what will the formula become? 6. What will the 10th term of this formula be? 7. What will represent the nth term of such a series? 8. What is the rule for finding any given term of such a series, when the first term and the ratio are known? 9. What is the rule for finding the sum of 5 terms of such a series? Multiply both members of this last equation by r, and we have sr — a r3 = ar+ar2+ar3tars Then, s—a and sr—a r5, being both equal to the same series of terms, are equal to each other. 8r-α p5 =8- α sr—8 = α pá (r-1) 8a5 α α We see that a 75 is the last term, art, multiplied by the ratio. What is the sum of 6 terms of the preceding series? Multiply the last term of the series by the ratio; from that product subtract the first term, and divide the remainder by the ratio, less one. The result will be equal to the sum of the series. 10 Find the sum of 5 terms of the series 60, 120, &c. .11 Find the sum of 6 terms of the series 45, 135, &c. 12 What is the sum of 10 terms of the series 1, x, x2, x3, &c.? This series may be written 1, x, xx, xx2 xx3, &c. PROPOSITION 1st.-If there be four terms in geometrical pro-gression, the second divided by the first, is equal to the fourth divided by the third. For, a, ar, a r2, a r3 may represent any series of the kind. Then, if the proposition be true Now, it is evident that this equation is true; for the first member reduced to its lowest terms, is r, and the second reduced to its lowest terms is r. PROPOSITION 2nd.-If three terms be in geometrical progression, the product of the first and last terms, is equal to the second power of the middle term. This is evident; for, if a, ar, a r2 represent these terms, the first term multiplied by the last, gives a2; and, the second power of the middle term is a2 2. PROPOSITION 3rd. The product of the first and last of 4 terms of a geometrical progression, is equal to the product of the two middle terms. How can you show that this proposition is true? 13. There are three numbers in geometrical progression; the middle one is 6, and the sum of the first and last, is equal to ten times the first term. What are the numbers? 14. There are four numbers in geometrical progression; the third is twice the second, and the sum of the numbers is 30. What are the numbers? 15. What will represent the sum of the series 1, r + 1, (r+ 1)2, (r + 1)3, &c., continued to the 20th term? Ans. (r + 1)2o —1 r 16. What will represent the sum of the series a, a (r † 1,) a (r+1), a (r+1), &c., continued to the 100th term? 17. A man drew the wine out of a full cask, and filled it with water. He then drew out half the mixture, and filled the cask again with water. This he continued until he had drawn out the fifth time, when there remained just 31 gallons of wine. How many gallons did the cask contain? SECTION LVI. On Indeterminate Equations. 1. Find two whole numbers whose sum is 5. 2. Find two whole numbers whose sum is 10. Find two numbers, such that the first added to twice the second, may equal 12. Assume y = 1, 2, 3, 4, 5, and we shall find x = 10, 8, 6, -4, 2. These are called indeterminate problems; because there are not sufficient data to determine the numbers; and, hence the questions admit of several answers. The number of answers, however, is limited, by the condition that they are to be in whole numbers; were it not for this condition, there would be an infinite number of answers to each of the preceding questions. 3. There are two whole numbers; one added to three times the other, makes 23. What are the numbers? 4. There are two whole numbers; one added to four times the other, makes 10. What are the numbers? ber. If it be added to a whole number, the sum must be a 3y 3 whole number. Now, is a whole number, by the condi 3y tions of the question. Hence +4 2y y 3 a whole number; and, if a whole number be subtracted from y 3 it, the remainder must be a whole number. Hence +4 y - = 3 must be a whole number. Put this equal to the least 2 and 3 are the numbers required. Will any other numbers answer the question? 6. Find two whole numbers, such that four times one added to three times the other, may equal 24. 7. Find two numbers, such that five times one and six times the other, may be equal to 50. 8. Find two numbers, such that three times one, times the other, may be equal to 26. 3x + 7y=26 and seven 3x=26-7y |