metrical ratio, there being 40 doors. What was the price of the house? Ans. $ 109951162777.50. 15. If the first term be 50, the ratio 1.06, and the number of terms 4, what is the sum of the series ? Ans. 218.7308. 16. A gentleman deposited annually $10 in a bank, from the time his son was born until he was 20 years of age. Required the amount of the deposits at 6 per cent., compound interest, when his son was 21 years old. Ans. $ 423.92,2+.. 17. If the first term be 7, the ratio 1, and the number of terms 5, what is the sum of the series ? Ans.-92 18. If one mill had been put at interest at the commence. ment of the Christian era, what would it amount to at com. pound interest, supposing the principal to have doubled itself every 12 years, January 1, 1837 ? Ans. $ 114179815416476790484662877555959610910619. 72.99,2. If this sum was all in dollars, it would take the present inhabitants of the globe more than 1,000,000 years to count it. If it was reduced to its value in pure gold, and was formed into a globe, it would be many million times larger than all the bodies that compose the solar system. PROBLEM III. To find the sum of the second powers of any number of terms, whose roots differ by unity. Rule. — Add one to the number of terms, and multiply this sum by the number of terms; then add one to twice the number of terms, and multiply this sum by the former product, and the last product, divided by 6, will give the sum of all the terms. 19. What is the sum of 10 terms of the series 12, 22, 32, 42, 52, 62, 72, 82, 92, 102 ? OPERATION. 10 x 10+ 1x 20+1 = 385 Ans. 20. What is the sum of 100 terms of the series 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, &c., to 1002 ? Ans. 338350. 21. Purchased 50 lots of land ; the first was one rod square, the second was two rods square, the third was three rods square, and so on, the last being 50 rods square. How many square rods were there in the 50 lots ? Ans. 42925. 22. Let it be required to find the number of cannon shot in a square pile, whose side is 80. Ans. 173880. Note. — A square pile is formed by continued horizontal courses of shot laid one above another, and these courses are squares, whose sides decrease by unity from the bottom of the pile to the top row, which is composed of only one shot. PROBLEM IV. To find the sum of the third power of any number of terms, whose roots differ by unity. RULE. — Add one to the number of terms, and multiply this sum by half the number of terms; the square of this product is the sum of all the series. 23. Required the sum of the following series :- 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123. OPERATION. 12 + 1= 13; 12 +2=6; 13x6=78; 78X 78= 6084 Ans. 24. I have 10 blocks of marble, each of which is an exact cube. A side of the first cube measures one foot, a side of the second 2 feet, a side of the third 3 feet, and so on to the 10th, whose side measures 10 feet. Required the number of cubical feet in the blocks ? Ans. 3025 cubic feet. 25. What is the sum of 50 terms of the series 13, 23, 33, 43, 53, 63, 73, &c., up to 503 ? Ans. 1625625. SECTION LXV. INFINITE SERIES. AN INFINITE SERIES is such as, being continued, would run on ad infinitum ; but the nature of its progression is such, that, by having a few of its terms given, the others to any extent may be known. Such are the following series : 1, 2, 4, 8, 16, 32, 64, 128, &c., ad infinitum. 125, 25, 5, 1, $ 25, T5, 25, &c., ad infinitum. To find the sum of a decreasing series. RULE. — Multiply the first term by the ratio, and divide the product by the ratio less 1, and the quotient is the sum of an infinite decreasing series. 1. What is the sum of the series 4, 1, 1, the old, &c., continued to an infinite number of terms ? OPERATION. 4 X 4 -=54 Answer. 268 DISCOUNT BY COMPOUND INTEREST. [SECT. LXVI. · 2. What is the sum of the series 5, 1, š, zł, &c., continued to infinity ? Ans. 61. 3. If the following series, 8, 4, 5, 7, &c., were carried to infinity, what would be its sum? Ans. 91. 4. What is the sum of the following series, carried to infinity: 1, 3, 5, zy, gft, &c. ? Ans. 11. 5. What is the sum of the following series, carried to infinity: 11, 4, 4s, &c. ? Ans. 12. 6. If the series , , , ib, z, &c., were carried to infinity, what would be its sum ? Ans. 11. Section LXVI. 1 1. What is the present worth of $ 600.00, due 3 years hence, at 6 per cent. compound interest ? OPERATION. 1.06) = 1.191016)600.00($ 503.77+ Ans. By analysis. — We find the amount of $1 at compound interest for 3 years to be $ 1.191016; therefore $ 1 is the present worth of $ 1.191016 due 3 years hence. And if $ 1 is the present worth of $ 1.191016, the present worth of 600 * 600 = 1 191016 = $ 503.77,1+ RULE. - Divide the debt by the amount of one dollar for the given time, and the quotient is the present worth, which, if subtracted from the debt, will leave the discount. 2. What is the present worth of $ 500.00, due 4 years hence, at 6 per cent. compound interest ? Ans. $ 396.04,6+. 3. What is the present worth of $1000.00, due 10 years hence, at 5 per cent. compound interest ? Ans. $ 613.91,3+. 4. What is the discount on $ 800.00, due 2 years hence, at 6 per cent. compound interest ? Ans. $ 88.00,34-. 5. What is the present worth of $ 1728, due 5 years hence, at 6 per cent. compound interest ? Ans. $ 1291.26. 6. What is the discount on $ 3700, due 10 years hence, at 5 per cent. discount, compound interest ? Ans. $ 1428.52. 7. What is the present worth of $ 7000, due 2 years hence, at 5 per cent. compound interest ? Ans. $ 6:349. 21. Section LXVII. ANNUITIES AT COMPOUND INTEREST. An annuity is a certain sum of money to be paid at regular periods, either for a limited time or for ever. The present worth or value of an annuity is that sum which, being improved at compound interest, will be sufficient to pay the annuity. The amount of an annuity is the compound interest of all the payments added to their sum. To find the amount of an annuity at compound interest. RULE. — Make $ 1.00 the first term of a geometrical series, and the amount of $1.00 at the given rate per cent. the ratio. 'Carry the series to so many terms as the number of years, and find its sum. Multiply the sum thus found by the given annuity, and the product will be the amount. EXAMPLES 1. What will an annuity of $60 per annum, payable yearly, amount to in 4 years, at 6 per cent. ? 2 3 4.374616 X 60 = $ 262.47,6+ Answer. Or, 1.06-1 . x 60=$ 262.47,6+ Answer. 2. What will an annuity of $500.00 amount to in 5 years, at 6 per cent. ? Ans. $ 2818.54,6+. 3. What will an annuity of $1000.00, payable yearly, amount to in 10 years ? Ans. $13180.79,47. · 4. What will an annuity of $ 30.00, payable yearly, amount to in 3 years ? Ans. $95.50,8+. To find the present worth of an annuity. As the first payment is made at the end of the year, its present worth or value is a sum that will amount in one year to that payment; and as the second payment is made at the end of the second year, its value is a sum that will, at compound interest, amount in two years to that payment; and the same principle is adopted for the third year, fourth year, &c. This may be illustrated in the following question. 270 ANNUITIES AT COMPOUND INTEREST. [SECT. LIVII. 5. What is the present worth of an annuity of $1.00, to continue 5 years, at compound interest ? The present worth of $ 1.00 for 1 year = $ 0.943396 $ 4.212363 By the above illustration, we perceive that the present worth of an annuity of $ 1, to continue 5 years, is $ 4.21,2+. Hence, having found the present worth of an annuity of $ 1 for any given time by Section LXVI., the present worth of any other sum may be found by multiplying it by the present worth of $1 for that time. RULE. — Multiply the present worth of the annuity of one dollar for the given time by the given annuity, and the product is the present worth required. Or, find the amount of the annuity by the last rule, and then find its present worth. 6. What is the present worth of an annuity of $ 60, to be continued 4 years, at compound interest ? First Method. $ 3.465104 Second Method. 1.06 – 1 ind =$ 4.374616 x .792093= $ 3.465102 x 60 = $ 207.90,6+ Ans. 7. A gentleman wishes to purchase an annuity, which shall afford him, at 6 per cent. compound interest, $ 500 a year for ten years. What sum must he deposit in the annuity office to produce it? Ans. $ 3680.04 +. 8. What is the present worth of an annuity of $ 1000, to continue 10 years? Ans. $ 7360.08. 9. What is the present worth of an annuity of $ 1728, to continue 3 years ? Ans. $ 4618.96. |