EXAMPLES FOR PRACTICE. 1. The first term is 5, the ratio 3, and the number of terms 6; what is the sum of the series? Ans. 1820. 2. The first term is 1024, the ratio, and the number of terms 8; what is the sum of the series? Ans. 2040. 3. The first term is 12, the ratio 3, and the number of terms 5; what is the sum of the series? 4. The first term is 102, the last term 4, and the ratio what is the sum of the series? Ans. 19.375. Ans. 1. Ans. 1. 5. The first term is 10, the ratio .5, and the number of terms 5; what is the sum of the series? 6. Find the sum of,, etc., to infinity. 7. Find the sum of 1,,, etc., to infinity. 8. Find the sum of 8, 4, 2, etc., to infinity. 9. Find the sum of 27, 9, 3, etc., to infinity. ANNUITIES. Art. 456. An annuity is a sum of money payable at equal intervals of time, such as annually, semi-annually, quarterly, &c. The beginning of an annuity is at a time before its first payment equal to one of its intervals. The end of an annuity is at the time of the last pay ment. The time between the beginning and the end of an annuity is called its time to run. The initial worth of an annuity is its value at beginning The present worth of an annuity is the sum of the present worths of the payments. The amount of an annuity is the sum of the interest amounts of the payments. It is equal to the interest amount of the present worth of the annuity. In reference to its beginning, an annuity is past, immediate, or deferred. A past annuity is an annuity which began in past time. An immediate annuity is an annuity which begins at the present time. Such an annuity is said to be in possession. A deferred annuity is an annuity which is to begin at a future time. If it is to begin after a certain interval, it is called a deferred annuity. If it is to begin after the occurrence of a future uncertain event, as the death of a certain person, it is called a reversionary annuity, or an annuity in reversion. A past annuity is said to be honored when its payments have been made; it is said to be dishonored, or forborne, or in arrears, when its payments have not been met. In reference to its time to run, an annuity is limited, or perpetual. A limited annuity is an annuity which has an end. A perpetual annuity, or an annuity in perpetuity, is an annuity which has no end. In reference to certainty, an annuity is either certain or contingent. A certain annuity is an annuity which is payable for a definite length of time. A contingent annuity is an annuity whose beginning or end is uncertain, because depending on an uncertain event, for example, the death of one or more persons. ANNUITIES AT SIMPLE INTEREST. CASE I. Art. 457. In an annuity certain at simple interest, to find the amount or final value. Ex. 1. If $300 are due annually, and the payments are withheld 4 years, what is then due, at 6% simple interest? Ans. $1680. EXPLANATION.-The first $300 amounts in 4 yr. to $372; the Rule.-Make the interest of one period the common differ- Art. 449. 2. What is the amount of an annuity of $500 for 8 years, 4. A worked for B one year, at $60 per month, payable 5. What is the amount of an annuity of $160 for 2 years, CASE II. Art. 459. To find the present worth of an annuity, Rule.-Divide the amount of the annuity by the amount of 1. What is the present worth of an annuity of $400 for 6 2. B leased a lot for 5 years for $1000 a year; what sum, Ans. $4142.85+. 3. What is the present worth of an annuity of $300 for ANNUITIES AT COMPOUND INTEREST. CASE I. Art. 459. To find the amount of an annuity at compound interest. Ex. 1. What is the amount of an annuity of $300 for 5 years, at 6% compound interest? Ans. $1691.13. EXPLANATION.-The fifth $300 is due without interest; the fourth amounts in 1 yr. to $300 × 1.06; the third in 2 yr. to $300 X 1.06 X 1.06 = $300 × 1.062; the second in 3 yr. to $300 × 1.06 X 1.06 X 1.06 = $300 X 1.063, &c. These numbers form a geometrical series, of which the annuity, $300, is the first term, the ratio is the amount of $1 for 1 yr., $1.06, the number of years is the number of terms, and the sum of the series is the amount of the annuity. Rule.-Make the amount of one unit of money for one period of time the ratio, the annuity the first term, and the number of periods the number of terms; then find its last term by Art. 454, and its sum by Art. 455. NOTE. The last term equals the annuity multiplied by the amount of $1 for the given time and rate per cent., taken from the table, Art. 385. 2. What is the amount of an annuity of $200 for 4 years, at 6% compound interest? Ans. $874.923. 3. What is the amount of an annuity of $300 for 25 years, at 6% compound interest? Ans. $16459.35. 4. A father deposited in a Savings Bank for his son, on his twelfth birthday $100, and the same amount on each subsequent birthday; how much did this amount to when the son was 21 years old, at 5% compound interest? Ans. $1257.77. 5. A man expends $200 annually for tobacco and drinks. If he should dispense with these and lend the $200, at 6% compound interest, how much would it amount to in 40 years? Ans. $30952.39. 6. Find the amount of an annuity of $500 for 5 years at 6% compound interest. CASE II. Art. 460. To find the present worth of an annuity, at compound interest. Rule.-Divide the amount of the annuity by the compound amount of $1 for the given time and rate per cent. taken from the table, Art. 385. 1. What is the present worth of an annuity of $400 for 6 years, at 6% compound interest? Ans. $1966.92+. 2. What is the present worth of an annuity of $500 for 10 years, at 5% compound interest? 3. An annuity of $300 for 20 years is in reversion 8 years. What is its present value, compound interest at 5%? Ans. $2530.47+. 4. An annuity of $200 for 15 years is in reversion 10 years. What is its present worth, at 6% compound interest? PERMUTATIONS. Art. 461. Permutations are changes in the arrangement of a given number of things. CASE I. Art. 462. To find the possible number of arrangements of a given number of things, taken all at a time. = The two letters b and c can be arranged, as usually written, in 1 X 2 2 ways; thus, bc and c b. The three letters b, c, and d can be arranged in 1 X 2 X 36 ways; thus, bed, bdc, cbd; cdb, dbc, dcb; that is, the new letter, d, takes three positions in each of the two arrangements b c and c b. A fourth letter, ƒ, can take four positions in each of the six arrangements of bcd; hence four things can make 1 × 2 × 3 X 4 24 arrangements. = Rule. Find the product of as many numbers of the natural series 1, 2, 3, &c., as there are objects to be arranged. |