2. The extremes are 1 and, and the ratio is ; what is the sum of all the terms? RULE.-Multiply the last term by the ratio, and divide the difference between the product and the first term by the difference between 1 and the ratio. 3. The extremes are 3 and 384, and the ratio is 2; what is the sum of all the terms? 4. The extremes are 4 and 8, and the ratio is ; what is the sum of all the terms? 5. What is the sum of all the terms of the infinite progression 8, 4, 2, 1, 1, 1, ....? The last term of this progression may be conceived as 0. 6. What is the sum of all the terms of the infinite progression 1,,, 27, 31, ? 7. What is the sum of 1+1+1+1, etc., to infinity? 8. The first is 7, the ratio 3, and the number of terms 4; what is the sum of all the terms? First find the last term by Art. 840. 9. A drover bought 10 cows, agreeing to pay $1 for the first, $2 for the second, $4 for the third, and so on; what did he pay for the 10 cows? 10. If a man were to buy 12 horses, paying 2 cents for the first horse, 6 cents for the second, and so on, what would they cost him? ANNUITIES 844. An Annuity is a sum of money payable annually. The term is also applied to a sum of money payable at any equal intervals of time. 845. A Certain Annuity is one which continues for a definite period of time. 846. A Perpetual Annuity, or Perpetuity is one which continues forever. 847. A Contingent Annuity is one which begins or ends, or both begins and ends, on the occurrence of some specified future event or events. 848. An Annuity Forborne, or in Arrears is one the payments of which were not made when due. 849. The Amount, or Final Value, of an annuity is the sum of all the payments increased by the interest of each payment from the time it becomes due until the annuity ceases. 850. The Present Worth of an annuity is such a sum of money as will, in the given time, and at the given rate per cent., amount to the final value. 851. An annuity is said to be deferred when it does. not begin until after a certain period of time; it is said to be reversionary when it does not begin until after the occurrence of some specified future event, as the death of a certain person; and it is said to be in possession when it has begun, or begins immediately. ANNUITIES AT SIMPLE INTEREST. 852. All problems in annuities at simple interest may be solved by combining the rules in Arithmetical Progression with those in Simple Interest. WRITTEN EXERCISES. 853. 1. What is the amount of an annuity of $300 for 5 years, at 6 per cent. simple interest? These sums form an arithmetical progression, in which the first term is the annuity, $300, the common difference is the interest of the annuity for 1 year, and the number of terms is the number of years. The sum of all the terms of this progression is $1680 (832), which is the amount of the annuity. 2. A father deposits annually for the benefit of his son, beginning with his tenth birthday, such a sum that on his 21st birthday the first deposit, at simple int., amounts to $210, and the sum due his son is $1860. Find the annual deposit, and at what rate per cent. it is deposited. OPERATION. 6 × (1st term + 210) = 1860. (832.) Hence, 1st term 310-210 = 100 = a. (210100) (12 − 1) = = 10 d. (830.) ÷ 0 ANALYSIS.-Here $210, the first deposit, is the last term; 12, the number of deposits, is the number of terms; and, $1860, the final value of the annuity, is the sum of all the terms. Using the principle of 832, we find the first term to be $100, which is the annual deposit. By 830, the common difference is found to be $10; hence 10 per cent. is the required rate. 3. What is the amount of an annuity of $150 for 51 years, payable quarterly, at 11 per cent. per quarter? 4. What is the present worth of an annuity of $300 for 5 years, at 6. per cent. ? 5. What is the present worth of an annuity of $500 for 10 years, at 10 per cent.? 6. In what time will an annual pension of $500 amount to $3450, at 6 per cent. simple interest? 7. Find the rate per cent. at which an annuity of $6000 will amount to $59760 in 8 years, at simple interest. 8. A man works for a farmer 1 yr. 6 mo., at $20 per month, payable monthly; and these wages remain unpaid until the expiration of the whole term of service. What is due the workman, allowing simple interest at 6 per cent. per annum ? ANNUITIES AT COMPOUND INTEREST. 854. All problems in annuities at compound interest may be solved by combining the rules in Geometrical Progression with those in Compound Interest. 1. What is the amount of an annuity of $300 for 5 years, at 6 per cent. compound interest? OPERATION. 300 x 1.065-300 .06 = 1691.13 ANALYSIS.-At the end of the 5th year the following sums are due: The 5th year's payment = $300, = $300 × 1.03, = The 4th year's payment + interest for 1 year $300 × 1.062, $300 × 1.063, $300 × 1.064. These sums form a geometrical progression, in which the first term is the annuity, $300, the ratio is the amount of $1 for 1 year, and the number of terms is the number of years. The sum of all the terms of this progression is $1691.13 (843), which is the amount of the annuity. 2. What is the present worth of an annuity of $300 for 5 years, at 6 per cent. compound interest? 3. Find the annuity whose amount for 25 years, at 6 per cent. compound interest, is $16459.35. 4. What is the present worth of an annuity of $700 for 7 years, at 6 per cent. compound interest? 5. An annuity of $200 for 12 years is in reversion 6 years. What is its present worth, compound interest at 6% ? 6. A man bought a tract of land for $4800, which was to be paid in installments of $600 a year; how much money, at 6 per cent. compound interest, would discharge the debt at the time of the purchase? 7. What is the present value of a reversionary lease of $100, commencing 14 years hence, and to continue 20 years, compound interest at 5 per cent. ? |