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RULE. Divide the first term by unity diminished by the rotio. Ex. 1. Find the sum of the infinite series

1+3+4+5+, etc.
Here

a=1, r=1.
Therefore S=I=11=2.
Ex. 2. Find the sum of the infinite series

1 1 1
Itatatout, etc.

Ans.
Ex. 3. Find the sum of the infinite series

1 1 1
1+ū+16+62+, etc.

. Ans.
Ex. 4. Find the sum of the infinite series

1 1 1
to +15+, etc.

Ans.

Ex. 5. Find the sum of the infinite series

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GUEST.–Give the rule for the sum of an infinite decreasing series. Explain the reason of the rule.

(218.) MISCELLANEOUS PROBLEMS.

Prob. 1. Find a number such that one third there. of, increased by one fourth of the same, shall be equal to one sixth of it, increased by 30.

Ans. 72. Prob. 2. Divide $1340 among three persons, A, B, and C, so that B may receive $100 more than A, and C $180 more than B. How much should each receive?

Ans. A $320, B $420, C $600. Prob. 3. In a mixture of wine and cider, one half of the whole, plus 21 gallons, was wine; and one third part, minus 6 gallons, was cider. How many gallons were there of each ?

Ans. 66 of wine and 24 of cider. Prob. 4. A's age is double of B's, and B's is triple of C's, and the sum of all their ages is 120 years. What is the age of each?

Ans. A's is 72, B's is 36, and C's is 12 years. Prob. 5. Two persons, A and B, lay out equal sums of money in trade; A gains $504, and B loses $348, and A's money is now double of B's. What sum did each lay out?

Ans. $1200. Prob. 6. A gentleman bought a chaise, horse, and harness for $315. The horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness. What did he give for each? Ans. $210 for the chaise, $70 for the horse, and

$35 for the harness.

Prob. 7. Two persons, A and B, have both the same income. A saves one fifth of his yearly; but B, by spending $500 per year more than A, at the end of 4 years finds himself $1000 in debt. What was his income?

. Ans. $1250. Prob. 8. A person in play lost one fourth of his money, and then won $3; after which he lost one third of what he then had, and then won $2. Lastly, he lost one seventh of what he then had, and found he had but $12 remaining. What had he at first?

Ans. $20. Prob. 9. A person goes to a tavern with a certain sum of money in his pocket, where he spends 8 shil. lings. He then borrows as much money as he had left, and going to another tavern, he there spends 8 shillings also. Then borrowing again as much money as was left, he went to a third tavern, where likewise he spent 8 shillings, and borrowed as much as he had left; and again spending 8 shillings at a fourth tavern, he then had nothing remaining. What had he at first?

Ans. 15 shillings. Prob. 10. A father gives to his five sons $950, which they are to divide according to their ages, se that each elder son shall receive $20 more than his next younger brother. What is the share of the youngest ?

Ans. $150. Prob. 11. A gentleman has two horses and two sad. dles, one of which cost $30, the other $5. If he places the best saddle upon the first horse, and the worst upon the second, then the latter is worth $5 more than the other. But if he puts the worse saddle upon the first horse, and the best upon the other, then the latter is worth twice as much as the first. What is the value of each horse ?

Ans. The first $50, the second $80. Prob. 12. There are two numbers whose sum is 37, and if three times the less be subtracted from four times the greater, and this difference be divided by 6, the quotient will be 6. What are the numbers ?

Ans. 16 and 21. Prob. 13. Find three numbers such that the first, with half the sum of the second and third, shall be 120; the second, with one fifth the difference of the third and first, shall be 70; and half the sum of the three numbers shall be 95.

Ans. 50, 65, and 75. Prob. 14. A banker has 2640 coins of two kinds, and there are four and a half times as many of one sort as of the other. How many has he of each sort?

Ans. 480 and 2160. Prob. 15. Divide the number a into two such parts chat one may be m times as great as the other.

а , та

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Prob. 16. How much money have I in my pocket when the fourth and fifth parts of the same together amount to $9?

Ans. $20. Prob. 17. Divide the number 46 into two parts, so that when the one is divided by 7, and the other by 3, the quotients together may amount to 10.

Ans. 28 and 18. Prob. 18. A fortress has a garrison of 2600 men, among whom there are 9 times as many foot soldiers, and 3 times as many artillery soldiers, as cavalry. How many of each corps are there?

Ans. 200 cavalry, 600 artillery, and 1800 foot. Prob. 19. I have a certain number in my thoughts, says A to B; try to guess it. I multiply it by 7, add 3 to the product, divide this by 2, subtract 4 from the quotient, and obtain 15. What number is it?

Ans. 5. Prob. 20. There are three numbers such that the second, divided by the first, gives 2 for a quotient and 1 for a remainder; while the third, divided by the second, gives 3 for a quotient with the remainder 3. The sum of these three numbers is 70. What are the numbers ?

Ans. 7, 15, and 48. Prob. 21. An arithmetician desires his scholars to find a number which he has in his mind from the following data. If, says he, you multiply the number by 5, subtract 24 from the product, divide the remainder by 6, and add 13 to the quotient, you will obtain this same number. What number is it?

Ans. 54. Prob. 22. Two purses together contain $300. If we take $30 out of the first and put it into the second, there will be the same sum in each. How much does each contain ?

Ans. The first $180, the second $120

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