The amount of this at the end of another year is 2a + ar+ 2 ar + a r2 or 2 a + 3 ar+ar3. Adding a to this it becomes 3a+3ar+ars. The amount of this for one year is 3 a +3ar+ar2 + 3 ar+3 ar2 + ar3, = a (3+6r+4r2 +r3). This is the amount at the end of the third year before the addition is made to the capital. The law is now sufficiently manifest. With a little alteration, the quantity 36r+4 r2 + r3 may be rendered the 4th power of 1+r. The three last coefficients are already right. If we add 1 to the quantity it becomes 4+6r+4 r2 +r3. Multiply this by r and it becomes 4r +6 r2+4r3 + r4. Add 1 again and it becomes 1+ 4r + 6 r2 + 4 r3 +rs. This is now the 4th power of 1+r, and it may be written (1+r)*. Subtract the 1 which was added last, and it becomes (1+r)4-1. Divide this by r, because it was multiplied by r and Subtract 1 again, because 1 was added previous to multiplying by r; and it becomes Substitute in place of the exponent 3, and multiply by a, and it becomes 2. A man deposited annually $50 in a bank from the time his son was born, until he was 20 years of age; and it was taken out, together with compound interest on each deposit at 3 per cent., when his son was 21 years of age, and given to him. How much did the son receive? 3. How much did the bankers gain by receiving the money, supposing they were able to employ it all the time at 6 per cent. compound interest? 4. A man has a son 7 years old, and he wishes to give him $2000 when he is 21 years old; how much must he deposit annually at 4 per cent. compound interest, to be able to do it? 5. If a man deposits in a bank annually $35, in how long a time will it amount to $500 at 6 per cent. compound interest? 6. The first slaves were brought into the American Colonies in the year 1685. Suppose the first number to have been 50, and that 50 had been brought each year for 100 years, and the rate of increase 3 per cent. How many would there have been in the country at the end of the hundred years? 1. A man died leaving a legacy to a friend in the following manner; a sum of money was to be put at interest, such that, the person drawing 10 dollars a year, at the end of 15 years the principal and interest should both be exhausted. What sum must be put at interest at 6 per cent. to fulfil the above condition? Let the learner generalize this example and form a rule; and then solve the following examples by it. 2. A man wishes to purchase an annuity which shall afford him $300 a year so long as he shall live. It is considered probable that he will live 30 years. What sum must he deposit in the annuity office to produce this sum, supposing he can be allowed 3 per cent. interest? N. B. The principal and interest must be exhausted at the end of 30 years. 3. If the man mentioned in the last example should die at the end of 18 years, how much would the annuity company gain? 4. If he were to live 43 years, how much would the company lose? 5. A man purchases an annuity for life, on the supposition that he shall live 45 years, for $15000, and is allowed 4 per cent. interest. How much must he draw annually that the whole may be exhausted? 6. A man has property to the amount of $35000, which yields him an income of 5 per cent. His annual expenses are $5000. How long will his property last him? 7. The number of slaves in the United States in 1810 was 1,191,000, and in 1820 the number was 1,531,000. What is the number at present, 1825, allowing the rate of increase to be the same? 8. There is a society established in the United States for the purpose of colonizing the free people of colour. Suppose the slaves to be emancipated as fast as this society can transport them away; how many must be sent away annually, that the number may be neither increased nor diminished? 9. How many must be sent away annually that the country may be cleared in 100 years? 10. If the colonization is not commenced till the year 1840, supposing the rate of increase to remain the same as from 1810 to 1820, how many must then be sent away annually, that the number remaining may continue the same? 11. How many must then be sent away annually, that the country may be cleared of them in 100 years? Miscellaneous Examples. 1. An express set out to travel 240 miles in 4 days, but in consequence of the badness of the roads he found that he must go 5 miles the second and 14 the fourth, less than the first. must he travel each day? day, 9 the third, How many miles Ans. 67, 62, 58, and 53 miles. |