We may perform this example by multiplication; thus, 4 X 3X 3X 3X3 X 3X 3X3 × 3 = $26244, Ans. But this process, you must be sensible, would be, in many cases, a very tedious one; let us see if we cannot abridge it, thereby making it easier. In the above process, we discover that 4 is multiplied by 3 eight times, one time less than the number of terms; consequently, the 8th power of the ratio 3, expressed thus, 38, multiplied by the first term, 4, will produce the last term. But, instead of raising 3 to the 8th power in this manner, we need only raise it to the 4th power, then multiply this 4th power into itself; for, in this way, we do, in fact, use the 3 eight times, raising the 3 to the same power as before; thus, 3481; then, 81 x 81=6561; this, multiplied by 4, the first term, gives $26244, the same result as before. A. 26244. Hence, When the First Term, Ratio, and Number of Terms, are given, to find the Last Term ;— Write down some of the leading powers of the ratio, with the numbers 1, 2, 3, &c., over them, being their several indices. Add together the most convenient indices to make an index less by 1 than the number of terms sought. Multiply together the powers, or numbers standing under those indices; and their product, multiplied by the first term, will be the term sought. 2. If the first term of a geometrical series be 4, and the ratio 3, what is the 11th term? Note. The pupil will notice that the series 3, 9, 27, 81, 243, powers. ( does not commence with the first term, but with 1, 2, 3, 4, 5, indices.) the ratio. The indices 5+3+2=10, and the powers under each, 243 x 27 x 9 = 59049; which, multiplied by the first term, 4, makes 236196, the 11th term, required. A. 236196. 3. The first term of a series, having 10 terms, is 4, and the ratio 3; what is the last term? A. 78732. 4. A sum of money is to be divided among 10 persons; the first to have $10, the second $30, and so on, in threefold proportion; what will the last have? A. $196830. 5. A boy purchased 18 oranges, on condition that he should pay only the price of the last, reckoning 1 cent for the first, 4 cents for the second, 16 cents for the third, and in that proportion for the whole; how much did he pay for them? A. $171798691,84. 6. What is the last term of a series having 18 terms, the first of which is 3, and the ratio 3? A. 387420489. 7. A butcher meets a drover, who has 24 oxen. The butcher inquires the price of them, and is answered, $60 per head; he immediately offers the drover $50 per head, and would take all. The drover says he will not take that; but, if he will give him what the last ox would come to, at 2 cents for the first, 4 cents for the second, and so on, doubling the price to the last, he might have the whole. What will the oxen amount to at that rate? A. $167772,16. 8. A man was to travel to a certain place in 4 days, and to travel at whatever rate he plansed; the first day he went 2 miles, the second 6 miles, and so on to the last, in a threefold ratio; how far did he travel the last day, and how far in all ? In this example, we may find the last term as before, or find it by adding each day's travel together, commencing with the first, and proceeding to the last, thus: 2+6+18+5480 miles, the whole distance travelled, and the last day's journey is 54 miles. But this mode of operation, in a long series, you must be sensible, would be very troublesome. Let us examine the nature of the series, and try to invent some shorter method of arriving at the same result. By examining the series 2, 6, 18, 54, we perceive that the last term (54), less 2 (the first term), 52, is 2 times as large as the sum of the remaining terms; for 2+6+18=26; that is, 54 - 252 ÷ 2 = 26; hence, if we produce another term, that is, multiply 54, the last term, by the ratio 3, making 162, we shall find the same true of this also; for 162-2 (the first term), = 160280, which we at first found to be the sum of the four remaining terms, thus, 2+6+18+5480. In both of these operations, it is curious to observe, that our divisor (2), each time, is 1 less than the ratio (3). Hence, When the Extremes and Ratio are given, to find the Sum of the Series, we have the following easy RULE. Multiply the last term by the ratio, from the product subtract the first term, and divide the remainder by the ratio, less 1; the quotient will be the sum of the series required. 9. If the extremes be 5 and 6400, and the ratio 6, what is the whole amount of the series? 10. A sum of money is to be divided among 10 persons in such a manner, that the first may have $10, the second $30, and so on, in threefold proportion; what will the last have, and what will the whole have? The pupil will recollect how he found the last term of the series by a foregoing rule; and, in all cases in which he is required to find the sum of the series, when the last term is not given, he must first find it by that rule, and then work for the sum of the series by the present rule. A. The last, $196830; and the whole, $295240. 11. A hosier sold 14 pair of stockings, the first at 4 cents, the second at 12 cents, and so on in geometrical progression; what did the last pair bring him, and what did the whole bring him? A. Last, $63772,92; whole, $95659,36. 12. A man bought a horse, and, by agreement, was to give a cent for the first nail, three for the second, &c.; there were four shoes, and in each shoe eight nails; what did the horse come to at that rate? A. $9265100944259,20. 13. At the marriage of a lady, one of the guests made her a present of a half-eagle, saying that he would double it on the first day of each succeeding month throughout the year, which he said would amount to something like $100; now, how much did his estimate differ from the true amount? A. $20375. 14. If our pious ancestors, who landed at Plymouth, A. D. 1620, being 101 in number, had increased so as to double their number in every 20 years, how great would have been their population at the end of the year 1840? ANNUITIES AT SIMPLE INTEREST. TXC. An annuity is a sum of money, payable every year, for a certain number of years, or forever. When the annuity is not paid at the time it becomes due, it is said to be in arrears. The sum of all the annuities, such as rents, pensions, &c., remaining unpaid, with the interest on each, for the time it has been due, is called the amount of the annuity. Hence, To find the Amount of an Annuity;— Calculate the interest on each annuity, for the time it has remained unpaid, and find its amount; then the sum of all these several amounts will be the amount required. 1. If the annual rent of a house, which is $200, remain unpaid (that is, in arrears) 8 years, what is the amount? In this example, the rent of the last (8th) year being paid when due, of course, there is no interest to be calculated on that year's rent. The amount of $200 for 7 years = $284 The amount of $200 The amount of $200 6 5 ...= $272 = $260 4.....$248 3.....$236 The amount of $200 2.....$224 The amount of $200..1............. = $212 $200 $1936, Ans. 2. If a man, having an annual pension of $60, receive no part of it till the expiration of 8 years, what is the amount then due? A. $580,80. 3. What would an annual salary of $600 amount to, which remains unpaid (or in arrears) for 2 years?-1236. For 3 years?-1908. For 4 years?-2616. For 7 years?-4956. For 8 years?-5808. For 10 years?-7620. Ans. $24144. 4. What is the present worth of an annuity of $600, to continue 4 years? The present worth (T LXVII.) is such a sum, as, if put at interest, would amount to the given annuity; hence, $600 $1,06 $566,037, present worth, 1st year. $600 $1,12 $535,714, $600 $1,18: = $508,474, $600 $1,24 = $483,870, 2d .... Ans., $2094,095, present worth required. Hence, To find the Present Worth of an Annuity;— Find the present worth of each year by itself, discounting from the time it becomes due, and the sum of all these present worths will be the answer. 5. What sum of ready money is equivalent to an annuity of $200, to continue 3 years, at 4 per cent.? A. $556,063. 6. What is the present worth of an annual salary of $800, to continue 2 years?-1469001. 3 years?-2146967. 5 years?-3407512. A. $7023,48. ANNUITIES AT COMPOUND INTEREST. TXCI. The amount of an annuity, at simple and compound interest, is the same, excepting the difference in interest. Hence, To find the Amount of an Annuity at Compound Interest ; Proceed as in ¶ XC., reckoning compound, instead of simple interest. 1. What will a salary of $200 amount to, which has remained unpaid for 3 years? The amount of $200 for 2 years = $224,72 The 3d year.. $200 00 A. $636,72 2. If the annual rent of a house, which is $150, remain in arrears for 3 years, what will be the amount due for that time? A. 477,54. Calculating the amount of the annuities in this manner, for a long period of years, would be tedious. This trouble will be prevented, by finding the amount of $1, or £1, annuity, at compound interest, for a number of years, as in the following TABLE I. Showing the amount of $1 or £1 annuity, at 6 per cent. compound interest, for any number of years, from 1 to 50. Yrs.6 per ct.Yrs.16 per ct.||Yrs.6 per ct.Yrs.16 per ct. Yrs. 16 per ct. 1 1,0000 11 14,9716 21 39,9927 31 84,8016 41165,0467 2 2,0600 2 16,8699 22 43,3922 32 90,8897 42 175,9495 3,1836 13 18,8821 23 46,9958 33 97,3431 43187,5064 3 4 4,3746 14 21,0150 24 50,8155 34 104,1837 44 199,7508 5 5,6371 1523,2759 25 54,8645 35 111,4347 45 212,7423 6 6.9753 16 25.6725 26 59,1563 36 119,120 46 226,5068 It is evident, that the amount of $2 annuity is 2 times as much as one of $1; and one of $3, 3 times as much. Hence, To find the Amount of an Annuity, at 6 per Cent. ;— Find, by the Table, the amount of $1, at the given rate and time, and multiply it by the given annuity, and the product will be the amount required. 3 What is the amount of an annuity of $120, which has remained unpaid 15 years? The amount of $1, by the Table, we find to be $23,2759; therefore, $23,2759 X 120 = $2793,108, Ans. 4. What will be the amount of an annual salary of $400, which has been in arrears 2 years?-824. 3 years ?-127344. 4 years?-174984. 6 years?-279012. 12 years?-674796. 20 years?-147142. Ans. $28099,56. 5. If you lay up $100 a year, from the time you are 21 years of age till you are 70, what will be the amount at compound interest? A. $26172,08. 6. What is the present worth of an annual pension of $120, which is to continue 3 years? In this example, the present worth is evidently that sum, which, at compound interest, would amount to as much as the amount of the given annuity for the three years. Finding the amount of $120 by the Table, as before, we have $382,032; then, if we divide $382,032 by the amount of $1, compound interest, for 3 years, the quotient will be the present worth. This is evident from the fact, that the quotient, multiplied by the amount of $1, will give the amount of $120, or, in other words, $382,032. The amount of $1 for 3 years at compound interest, is $1,19101; then, $382,032 $1,19101 = $320,763, Ans. this Hence, To find the Present Worth of an Annuity ;— Find its amount in arrears for the whole time; amount, divided by the amount of $1 for said time, will be the present worth required. Note. The amount of $1 may be found ready calculated in the Table of compound interest, ¶ LXXI. 7. What is the present worth of an annual rent of $200, to continue 5 years: A. $842,472. The operations in this rule may be much shortened by calculating the present worth of $1 for a number of years, as in the following TABLE II. Showing the present worth of $1 or £1 annuity, at 6 per cent. compound interest, for any number of years, from 1 to 32. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. Years. 6 per cent. |