greatest. For the divisor is the product of the multiplier by r-1. Cor. 2.-If the common ratio be 2, the difference of the extremes is the sum of all the terms except the greatest. Cor. 3.-If a descending series be interminate, the least term may be considered = 0, and the sum = yr 1. Required the 8th term of the series 4, 8, 16, &c. 4X274 x 128 = 512. 2. Required the sum of 12 terms of the series 1, 3, 9, 312-1 531441-1 27, &c. =265720. 3 = 2 3. Required the sum of 8 terms of the series 1, }, }, 4. Given the extremes a and y, and the sum of the series s, to find the common ratio and the number of terms. s—a Ans. r== Having found r, -1. And in logarithms, where R, Y, and A represent the logarithms of Y-A+R r, y, and a, (n-1)RY—A, and n = R QUESTIONS. 1. To find four numbers in arithmetical progression, such, that their sum shall be 56, and the sum of their squares 864. Let the series be x, x+y, x+2y, x+3y, their sum 4x+6y 56, or 2x + 3y 28, and the sum of their squares 4x2+12xy +14y2864, from which subtract 2x+3y|2 = 282, or 4x2 + 12xy + 9y2 784; the remainder gives 5y80, or y=4, and x = 8; and the numbers are 8, 12, 16, 20. 2. To find three numbers in arithmetical progression, such, that their sum shall be 9, and the sum of their cubes 153. Let the numbers be x- -y, x, x+y, their sum 3x9, the sum of their cubes 3x+6xy2 = 153. Ans. The numbers are, 1, 3, 5. 3. To find three numbers in arithmetical progression, such, that their sum shall be 15, and the sum of the squares of the extremes 58. The numbers, x -y, x, x+y. Ans. 3, 5, 7. 4. To find four numbers in arithmetical progression, such, that the sum of the extremes shall be 8, and the product of the means 15. Ans. 1, 3, 5, 7. 5. To find four numbers in arithmetical progression, such, that the sum of the squares of the means shall be 52, and the sum of the squares of the extremes 68. Ans. 2, 4, 6, 8. 6. A traveller goes 9 miles a-day: after 7 days another sets out after him, and travels 4 miles the first day, 5 miles the second, 6 miles the third, and so on. In what time will he overtake the first? 8+x-1 x = (x+7)9. Ans. 18 days. 7. To find three numbers in geometrical progression, such, that their sum shall be 7, and the sum of their squares 21. Let x, y, z, be the numbers. xz=y2, x+y+ z = 7, x2 + y2+z2 = 21. Ans. 1, 2, 4. 8. To find four numbers in geometrical progression, such, that their sum shall be 30, and that the greatest shall be equal to the sum of the means multiplied by 13. x, xy, xy2, xy3, the numbers. Ans. 2, 4, 8, 16. 9. To find three numbers in geometrical progression, such, that their product shall be 64, and the sum of their cubes 584. x, xy, xy2, the numbers. x5y364, x3 × (1+y3 +y)=584. Ans. 2, 4, 8. pro 10. To find three numbers in geometrical progression, such, that the sum of the first and third shall be 52, and their duct 100. Ans. 2, 10, 50. 11. To find two mean proportionals between 4 and 256. 4, 4x, 4x2, 256, are the proportionals. Ans. x5 = 256 64, x=4, the numbers 4, 16, 64, 256. 12. Given the sum of the squares a, of three numbers in arithmetical progression, and the excess of the square of the mean above the product of the extremes b; to find the numbers. 13. Given the product of the extremes a, and the product of the means b, of four numbers in arithmetical progression; to find the numbers. 14. Given the number of terms n, of an arithmetical pro gression, their sum a, and the sum of their squares b; to find the terms. 15. Suppose two travellers set out at the same time from two places of which the distance is given, p. The miles travelled by the first per day form a decreasing arithmetical progression, of which the first term is given, a, and the common difference d. Those travelled by the second form an increasing series, of which the first term is b, and the common difference c. In what time will they meet? Let a+b=m, and с 1 m d=n. 2p m Ans. —±√(+(−1)2), or (if n = 0); }; 2 n n n m 16. Given the sum s of five numbers in geometrical progression, and the sum of their squares a; to find the numbers. Suppose v sum of the first and third, then v= a —and 28 IN SIMPLE INTEREST, the interest is computed on the prin cipal only. Let p = principal or money lent, t = time, r = rate or interest of £1 for the time one, i = interest for the whole time, a = amount or sum of principal and interest; then rt = interest of £1 for the time t, and 1+rt the amount of £1, and p× (1+rt)=p+prt=p+ia the amount of the whole; from which equations the value of any of the quantities concerned may be found in terms of the others. In COMPOUND INTEREST, the interest at each term of payment is added to the principal, and the amount is the principal for the next term. Let R=1+r the amount of £1 for the first term, it will be the principal for the next term, and the interest upon it will be Rr, and the amount Rr+R=R(r+1)=R2 will be the principal for the next term. In like manner we find that the amounts at the end of the following terms will be R3, R4, &c.; and at the end of the time t it will be Rt, and for the principal p it will be PRt the amount, and the interest will be pR-pia—? from which equations any of the quantities may be expressed in terms of the rest. I annual meet the gran ne, men n3 — " = pares this princa for the ime = vach vill mersure he De amount of an amury of £ dr hat time. But n = 3, and and or any mauity *. therefore the amount vil be it will be And f be equal to the present 3 value of this amuiry, then "'—*= p3°, and p=*—*, OF REVERSINs-When the annuity does not cemmence till some time afer this, it is said to be in reversion. The amocat, f it were to commence just now, would but if it commence a years after this, it will be nx R be R$ R = 4, and the present worth p = 1; × _— No. From these equations any of the quantities may be expressed in terms of the others. 1 IN A FREEHOLD ESTATE, the value y = when the rent is £1, and it commences just now; and is its value, when 1 Rr it does not commence till s years after this, y is called the year's purchase or perpetuity, and ay the value of the estate, of which the rent is a, and is the value in reversion. R$ ANNUITIES ON LIVES.-Adopting Mr De Moivre's hypothesis, that of a certain number born at one time, one dies every year until the whole is extinct, a supposition which agrees nearly with observation, for ages between 10 and 60. An annuity of £1 for a given life will be the sum of the n- -1 n-2 n-3 series n is the complement of the age, or what it wants of the age at which the oldest dies, which he supposed to be 86, and r the amount of £1 for a year. worth of an annuity of £1 for n— -1 years. Again, the value of an annuity for two joint lives, of which the complements are n and m, (the greatest m) value of the oldest life, the value of the two lives is (n − 1)p — s × (2p+1—(mn)), where p = perpetuity. m If a question occur which involves both interest and annuities, an equation may be found answering to it by comparing with one another the values of the quantities found separately. 1. What will £1000 amount to in 10 years, at 5 per cent. compound interest? Ans. £1628, 16s. 2. What principal will, in 15 years, amount to £2000, at per cent. compound interest? Ans. £1110, 12s. 3. In what time will £200 amount to £318, 16s., at 6 per cent. compound interest? Ans. 8 years. 4. In what time will a sum of money double itself, at 4 per cent. compound interest? 5 1.042. Ans. 17-6 years. 5. Required the amount of £20 annuity for 40 years, at Ans. £2536, 16s. per cent. 6. What annuity will, in 7 years, amount to £79, at 4 per cent.? Ans. £10. 7. What is the value of an annuity of £20, for a life of 54 years of age, at 4 per cent. ? Ans. £209.56. 8. What is the value of an annuity of £20, during the joint lives of two persons, whose ages are 35 and 25 years, at 4 per cent. ? Ans. £2219. 9. When 12 years of a lease of 21 years were expired, a renewal for the same term was granted for £1000.* Eight years of that lease are now expired, and it is required what sum should be paid for a corresponding renewal of the lease, reckoning 5 per cent. compound interest. From the first transaction, find the annuity n = £175.029, and from it find p, the present worth of the annuity in reversion £599.93. |