geometrical progression, when the first term, the number of terms, and the ratio, are given; this has been done by case V., under geometrical progression. The rule may be stated as follows: RULE. From the amount of $1, for 1 year, raised to a power whose exponent is equal to the number of years, subtract $1, divide the remainder by the interest of $1, for 1 year, then multiply the quotient by the annuity. NOTE. The different powers of the amount of $1, for one year, may be taken from the table under Art. 69. Examples. 1. What is the amount of an annuity of $200, which has been forborne 14 years, at 6 per cent., compound interest? From table under Art. 69, we find the 14th power of the amount of $1, for one year, at 6 per cent., to be $2.260904; subtracting $1, and dividing the remainder by $0.06, the interest of $1 for one year, we get $21.01506, which, multiplied by $200, the annuity gives $4203.012, for the amount required. 2. Suppose a person, who has a salary of $700 a year, payable quarterly, to allow it to remain unpaid for 4 years. How much would be due him, allowing quarterly compound interest at 12 per cent. per annum? Ans. $3527.454. 3. What is due on a pension of $150 a year, payable halfyearly, but forborne 2 years, allowing half-yearly compound interest, at 6 per cent. per annum? Ans. $313.772. 4. What is due on a pension of $350 a year, payable quarterly, but forborne 2 years, allowing quarterly compound interest, at 12 per cent.? Questions under this rule may be easily wrought by the following table. This table shows the amount of an annuity forborne for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent, compound interest. 4 5 6 7 4.310125 4.374616 5.525631 5.637093 4.183627 4.246464 30 5. What is due on a pension of $1000, which has been forborne 27 years, at 3 per cent., compound interest? From the above table we find the amount of an annuity of $1, for 27 years, at 3 per cent., to be $40.709634, which, multiplied by $1000, gives $40709.634, for the amount due. 6. What is the amount of an annuity of $50, which has been forborne 30 years, at 6 per cent., compound interest? 7. What is the amount of a pension of $300, which has been forborne 19 years, at 5 per cent., compound interest? 8. What is the amount of a pension of $900, which has been forborne 17 years, at 4 per cent., compound interest? 9. What is the amount of an annuity of $75, which has been forborne 13 years, at 5 per cent., compound interest? CASE II. To find the present worth of an annuity which is to terminate in a given number of years. The present worth of an annuity, is obviously such a sum of money as will, at compound interest, produce an amount equal to the amount of the annuity. Therefore, if we find the amount of the annuity by Case I., we may consider it as the amount of a certain principal, which principal is the same as the present worth. We have already been taught how to find the present worth, by rule under compound discount. Hence, we have this RULE. First, find the amount of the annuity, as if it were in arrears for the whole time, by the aid of the table under Case I., of ANNUITIES. Then, find the present worth of this amount for the given time and rate per cent., by the use of the table under coмPOUND DIS COUNT. Examples. 1. What is the present worth of an annuity of $500, to continue 10 years, interest being 6 per cent. ? By the table under Case I., of annuities, we find the amount of an annuity of $1, for 10 years, at 6 per cent., to be $13. 180795, this, multiplied by 500, gives $6590.3975, for the amount of the annuity. Now, by the table under compound discount, we find the present worth of $1, for 10 years, at 6 per cent,, to be $0.558395, which, multiplied by 6590.3975, gives $3680.045, for the present worth required. 2. What is the present worth of an annuity of $100, to continue 20 years, at 5 per cent. interest? Ans. $1246.222. The work under this Rule may be very much simplified by the use of the following table. The following table gives the present worth of an annuity of $1, or £1, for any number of years, not exceeding 30, at 3, 4, 5, and 6 per cent. 9.294984 9.712249 14 11.296070 10.563122 9.898641 |