of 3%, the total interest for a year is $1500. (0.03)= $45. The total sum invested at the end of a year would be $1545. Let, in general, P represent a sum of money in dollars. Let r represent a yearly rate of interest. Then Pr represents the yearly interest on P, and P+PrP(r+1) represents the total investment, principal and interest, at the end of a year. Similarly, P(r + 1)r is the second year's interest, and P(r + 1)r + P(r + 1) = P (r2 + r + r + 1) = P (r + 1)2 is the total investment at the end of two years. In general, A = P(r + 1)" (1) is the total accumulation at the end of n years. If we know r, P, and n, we can by (1) find A. If we take the logarithm of both sides of the equation, we have or log Alog P + n log (r+1), n = log A-log P log (r+1) Hence if we know A, P, and r, we can find n. (2) If the interest is computed semiannually, we have as interest r at the end of a half-year P., while the entire sum would be P 2 + 1). Reasoning as above, we find that if the interest is com puted semiannually, the accumulation at the end of ʼn years is If the interest is computed k times a year, we have at the end In such exercises as the following, four-place tables are not sufficiently exact to obtain perfect accuracy. In general, the longer the term of years and the more frequent the compounding of interest, the greater the inaccuracy. 1. If $1600 is placed at 31% interest computed semiannually for 13 years, to how much will it amount in that time? 2. After how long will $600 at 6% computed annually amount to $1000 ? In the following exercises the interest is computed annually unless the contrary is stated. 3. To what will $3750 amount in 20 years if left at 5% interest? 4. To what sum will $25,300 amount in 10 years if left at 5% interest computed semiannually? 5. To what does $1000 amount in 10 years if left at 6% interest computed .(1) annually, (2) semiannually, (3) quarterly, (4) monthly? 6. A sum of money is left 22 years at 4% and amounts to $17,000. How much was originally put at interest? 7. What sum of money left at 41% for 30 years amounts to $30,000 ? 8. What sum of money left 10 years at 41% amounts to the same sum as $8549 left 7 years at 5%? 9. If a man left a certain sum 11 years at 4%, it would amount to $97 less than if he had left the same sum 9 years at 5%. What was the sum? 10. Which yields more, a sum left 10 years at 4% or 4 years at 10%? What is the difference for $1000? 11. Two sums of money, $25,795 in all, are left 20 years at 43%. The difference in the sums to which they amount is $14,660. What were the sums? 12. At what per cent interest must $15,000 be left in order to amount to $60,000 in 32 years? 13. At what per cent must $3333 be left so that in 24 years it will amount to $10,000? 14. Two sums of which the second is double the first but is left at 2% less interest amount in 36 years to equal sums. At what per cent interest was each left? 15. In how many years will a sum double if left at 5% interest? 16. In how many years will a sum double if left at 6% interest computed semiannually? 17. In how many years will a sum amount to ten times itself if left at 4% interest? 18. In how many years will $17,000 left at 41% interest amount to the same as $7000 left at 51% for 20 years? 19. On July 1, 1850, the sum of $1000 was left at 41% interest. When paid back it amounted to $2222. When did this occur? 20. Prove formulas (1), (3), and (5) by complete induction. where a, b, g, are real numbers, is called a continued fraction. We shall consider only those continued fractions in which the numerators b, d, f, etc., are equal to unity and in which the letters represent integers, as for example tion is said to be terminating. When the fraction is not terminating it is infinite. We shall see that the character of the numbers represented by terminating fractions differs widely from that of the numbers represented by infinite continued fractions. We shall find, in fact, that any root of a linear equation in one variable, i.e. any rational number, may be represented by a terminating continued fraction, and conversely; furthermore, that any real irrational root of a quadratic equation may be represented by the simplest type of infinite continued fractions, and conversely. 219. Terminating continued fractions. If we have a terminating continued fraction, where a1, a2, are integers, it is evident that by reducing to its simplest form we obtain a rational number. The converse is also true, as we can prove in the following THEOREM. Any rational number may be expressed as a terminating fraction. Letrepresent a rational number. Divide a by 6, and let a, be the quotient and e (which must be less than 6) the remainder. Then (§ 26) a b 1 с + / с Divide b by c, letting a, be the quotient and d (which must be less than c) the remainder. Then Continuing this process, the maximum limit of the remainders in the successive divisions becomes smaller as we go on, until finally the remainder is zero. Hence the fraction is terminating. It is noted that the successive quotients are the denominators in the continued fraction. EXERCISES 1. Convert the following into continued fractions (a) 77. |