Arithmetic series be divided by a number less by one than the number of terms, the result will be the common difference. This divisor being greater by one than the number of means, the common difference may be found by dividing the difference between the first and last terms by the number of means increased by 1.

Cor. 3. Hence is apparent the method of inserting Arithmetic means between two given numbers; for the common difference being found as above, the successive terms may be readily formed from the first by the Prop.

Prop. 80.--To find any required term in a Geometric series from the first term, and the common ratio.

In a Geometric series the ratio of any term to the preceding is the same, so that any term may be formed from the preceding by multiplying by the same number, which is called the common ratio. Hence in the second term the common ratio will enter as a factor once; in the third term twice; in the fourth term three times; and so on. Whence it appears that the common ratio will appear as a factor of any term a number of times less by one than the number of the term.

Cor. 1. Hence the last term is equal to the product of the first term multiplied by a power of the common ratio of a degree less by one than the number of terms.

Cor. 2. Hence if the first and last terms be given of a Geometric series, the quotient of the first by the last will be a power of the common ratio of a degree less by one than the uumber of terms; whence the common ratio may be found.

Cor. 3. Hence if it be required to insert any number of Geometric means between two numbers, the common ratio being determined, the terms themselves may be found by the Prop.

Prop. 81. To find the sum of a Geometric series.

Let the terms of the series be multiplied by the common ratio; the result will be a series containing all the terms of the former except the first, and another term, which is equal to the product of the last term by the common ratio; also it is manifest, that the sum of this series is equal to the sum of the former multiplied by the common ratio. Hence the difference between these series is equal to the difference between the first term, and the last multiplied by the common ratio; but it is also equal to the difference between the sum of the first and the same sum multiplied by the common ratio, i.e. to the product of the sum of the first multiplied by the difference between 1 and the common ratio. Hence the sum may be obtained by

R

dividing the difference between the first term and the last multiplied by the common ratio, by the difference between 1 and the common ratio.

Prop. 82.-The powers of a number, greater than unity, increase, while those of a number, less than unity, decrease continually without limit.

Any number repeated more than once is greater than itself; but if a number greater than unity be multiplied by itself, it will be repeated more than once; therefore the square of such a number is greater than the first power; the cube is greater than the square, &c. The powers therefore of a number greater than unity increase continually. And this increase has no limit. For if a number however small be continually added to another, there will in time result a number greater than any other, which may be assigned. Much more therefore, when the numbers successively added are continually increased, will this be the case. But in forming the cube from the square, a greater number is added than in forming the square from the first power, since in the former case a number greater than the square is added; and in the latter, a number less than the square. Similarly it may be shown, that in forming all the successive powers of a number greater than unity, numbers are added which continually increase. Hence by raising the number to some power, a number may he formed greater than any, which may be assigned.

Again: a part, or parts of a number less than the whole number of parts in it, is less than the whole. But if a number be multiplied by a proper fraction, it is divided into a number of parts equal to the denominator, and a number of these parts are taken equal to the numerator, and therefore less than the denominator. Therefore the product is less than the multiplicand. Hence the square, cube, fourth powers, &c. of a proper fraction, are respectively less than the first, second, third, &c. powers, or the powers of a proper fraction continually decrease. And this decrease has no limit. For a proper fraction is equal to unity divided by the same fraction inverted, which is greater than unity; and which therefore can be made as large as we please, by raising it to some power. By raising the proper fraction therefore to some power, a result may be obtained equal to unity divided by a number as large as we please, i.e. such a part of unity, as that any number of these parts we please are required to form a quantity as large as unity, that is, as small a part of unity as we please. Hence the powers of a proper fraction may be made as small as we please, or decrease continually without limit.

Prop. 83.-To find the limit of the sum of an infinite decreasing Geometric series.

The limit of the sum of a series is a number, to which the sum of a finite number of terms becomes more and more nearly equal, as the number of terms is increased; but to which the sum is never actually equal, however many terms be taken.

The algebraical expression for the sum of a geometric series, whose first term is and common ratio which =

a,

r, is

a

1.

arn

- r 1 -p

Now if r be a proper fraction, r" may be made as small as we please, and

arn

therefore - may be made as small as we please, by increasing n. Hence

1 -r

α

•

the sum of an infinite decreasing geometric series continually approximates, though no finite number of terms can ever equal, the quantity Therefore the limit of the sum of such a series is the ratio of the first term to the difference between 1 and the common difference.

Prop. 84.-To prove and explain the Rule for finding the time at which several sums due at different times may be paid together.

The principle on which the ordinary rule is founded is the following. It is said that the debtor, by holding the debts in his hands, gains the advantage of the interest on them for the several times, for which he is allowed to hold them; and that therefore, if he pay them all at one time, this time should be determined by the condition that he should still gain the same aggregate advantage as before. In other words, it is said that the sum of the interest on the several debts for the times, at which they are due, ought to be equal to the interest on the sum of the debts for the equated time. This implies that the sum of the interest on the debts paid after they are due should be equal to the sum of the interest on the debts paid before they are due, for the difference of the times at which they are, and ought to be, paid; and also that the sum of the amounts of the several debts for the difference between the longest time and their own times, should be equal to the amount of the sum of the debts for the difference between the same time and the equated time. On these principles, if £ S1, £ S2, £ S3, &c. be several sums due at times T1, T2, T3, &c. terms respectively, and r be the interest on £1 for one term, and t the number of terms in the equated time, we have

{S, T1 + S2 T2 + S, T3+ &c. }=r {$1 + S2+$3 + &c. }

2 2

whence t =

S1 T1 S2 T2+ S3 T3 + &c.

2

S1 + S2+ S3+ &c.

which formula expresses the ordinary rule.

t

But with reference to these principles, it may be said, that the debtor by paying at the equated time gains the full advantage of leaving the debts unpaid their own times, sooner than he would in this case do. And again, by paying a debt before it is due, the debtor loses not the interest but the discount, and therefore the interest on the debts paid after they are due, should be equal to the discount on those paid before they are due, which would give the equated time smaller than the above formula. But the ordinary Rule is sufficiently exact for practical use. See note.

Prop. 85.-To prove the Rule for finding the amount of an annuity.

If A be the number of pounds in the annuity, n the number of terms for which it is to be paid, r the interest on £1 for 1 term, then the amount of the 1st payment, which is forborne for n 1 years is A (1+r)-1; that of the second payment is A (1+r)-2; and so on; that of the last is A. Hence the sum of the amounts.

=

= A { (1 + r) n − 1 + (1 + r)x−2 +&c. +1}

which expresses the Rule, (1+r)n being the amount of £1 at Compound interest.

Prop. 86.-To prove the Rule for finding the present value of an annuity.

Let P be the number of pounds in the sum, which, put out to interest, will be sufficient to pay the annuity A for n years. Then

nth year = P(1+r)¤ — 1—A(1+r)n − 2 —A(1+r)¤ — 1—&c.—A the amount of this last should be just sufficient to pay the nth annuity; P (1 + r)" — A (1 + r)n − 1 — A (1 + r)n − 2 — &c. — A = 0

or

which expresses the Rule, and shows that P is such a sum, as put out at Compound Interest would amouut to the amount of the annuity.

to; so that is the limit of the value of the annuity, as the number of

years for which it is paid is indefinitely increased. Hence if the annuity be

A

perpetual P -.

Cor. 2. This Prop. enables us to find the value of a leasehold, or the fine payable for a lease of any number of years, this being the value of an annuity equal to the difference between the rack-rent and the lease-rent. Also Cor. 1 gives the value of a freehold, which is a perpetual annuity.

Prop. 87.-To prove the Rule for finding the value of a deferred annuity.

The value of an annuity to commence after m years, and to continue n years, is evidently the difference between the values of the annuity for m+n years, and for m years. Hence if P be the present value,

Cor. This Prop. enables us to find the value of Reversionary Annuities.

Prop. 88. To prove the Rule for finding the annuity, which can be purchased for a given sum.

By Prop. 86 it appears, that the present value varies as the annuity, and hence the purchasable annuity varies as the sum invested. Therefore if £A be the annuity, which may be purchased with a sum, £ P, and £ P1 be the present value of £1 annuity,

Prop. 89.-To explain the methods of working questions in

Exchanges.

1st. Let it be required to determine how much coin of one country is equivalent to a given amount of another, the course of exchange between the