steps, I will repeat it here in the numbers of the above series, which will enable us to make the full comparison of its general result with any individual case that may The series chosen gives the following numbers in the first proportion, under the supposition of the number of terms n being 7.

Occur.

Remark. I here permitted the quantity to be subtracted to be the greater, both in the numerator and in the denominator; this, though apparently a contradiction, is compensating on the same ground as has been shown above that the objects themselves disappear in a rule of three, when they appear equally, both in numerator and in denominator; the result here is therefore equally positive. The signs of addition or subtraction, that is, +, and, compensate as equal quantities in numerator and in denominator, exactly like the quantities themselves. It will easily be seen, that if the series had been a decreasing one, the case would have been the reverse; the ratio being in that case a fraction, the numerator and the denominator would both have presented positive numbers, that is, the subtracting quantities, being fractions, would both be smaller than the unit.

The above expression for the value of the sum of

a geometric progression is therefore the rule (to express it in the common language of arithmetic) by which this sum is to be calculated. It can be stated very simply thus:

Take the difference between unity and the constant ratio elevated to the power indicated by the number of terms, divide this by the difference between unity and the constant ratio, and multiply the quotient by the first term.

This rule is evidently adapted both to increasing and to decreasing geometrical progressions.

$108. The foregoing expression, or formula, again presents us four quantities mutually depending upon each other, in the manner expressed by it; we may therefore conclude: that any three of them given determine the fourth; which might form as many distinct problems, as shown in the arithmetic series; we will here only show how to find the first term, the other parts being given.

The last step of the reduction of the proportion evidently gives:

or, in words: Divide the difference between unity and the constant ratio, by the difference between unity and the ratio elevated to the power indicated by the number of terms, and multiply the quotient by the sum of the series.

To determine the constant ratio, or the number of the term, when the other parts are given, requires more extensive deductions and calculation than the plan of these elements admits of; the first requires the solution of what is called a higher equation, and the second the use of logarithms, which both lie beyond our present limits.

CHAPTER III.

Of Compound Interest.-Idea of Annuities.

109. We have seen in its proper place, that the calculation of simple interest was a simple multiplication of the capital by the decimal fraction representing the interest per hundred, and in the Compound Rule of Three the other questions have been treated which relate to this subject. But, as well for the transactions of monied institutions, as for various other calculations, in political economy and otherwise, the interest after the year, or any other term agreed upon, is considered as again bearing interest, and thus the interest increases at the same rate as the capital itself. This introduces of course a mode of calculation completely different, and partaking of the nature of the Progressions: its principles shall here be treated separately, and with the addition of payments at determined terms, as the interests or annual payments, called annuities, of which it may be proper here to give only the first principles, without going into the details which more intricate speculations introduce into them, as they would draw us out of our prescribed limits.

We shall take the liberty of making use of letters to designate the quantities, until we give them actual values, by way of example; in order to give to the reasoning that general form which it is so advantageous to introduce in the higher branches of arithmetic. Thus we will call the capital and the rate of the per centage = r; and proceed with these as if they were known numbers, indicating the operations by means of the signs which we have long been familiar with.

=

C,

The capital having been one year at interest, it

will be worth, together with that interest, CrC C (1+r)

(for the C multiplies the unit and the rate per cent. r.) This being now the capital on interest for the second year, it will produce an interest = C(1+r) r; and the whole value of the capital and interest at the beginning of the third year will be the sum of the last year's capital and the interest of the same, namely:

=

C(1+r) + C(1+r) r = C(1+r) (1+r) C(1+r)2 (for here the C (1+r) is again a multiplier for the unit and the rate per cent. = r, and so in each following year.) This capital, at the same interest, in the third year will produce an interest =

1

C. r (1+r)2

which added to the last capital, gives at the beginning of the fourth year the value of

C(1+r)2+ Cr (1+r)2 = C(1+r)2(1+r) = C(1+r)2

This is therefore the law of the increase of a capital put out upon compound interest; which for any number of years, say n, would give

C(1+r)" 8; or,

In order to obtain the value of the whole capital at the end of the last year, the rate of interest added to unity, raised to the power indicated by the number of years elapsed, is to be multiplied into the original capital.

To show the same operation in numbers, let us suppose a capital, C= 7500, at the rate of 6 per cent. compound interest; this (expressing the per centage in a decimal fraction) evidently gives:

The first year's interest:

7500 × 0,06

The capital at the end of the first year :

75007500 X 0, 06

which will be more easily calculated thus:

16

7500 × 1,06

The second year's interest will be ;

7500 X 1,6 X 0,06

The capital at the end of the second year : 7500 X 1, 16 + 7500 X 1,06 ×0, 06

or, again expressed more simply:

7500 X 1,06 × 1,06 = 7500 (1,06)2

It will progress in this manner every year by the power of 1,06; that is, the original capital will be multiplied by 1,06 in continued multiplication of as many factors as the number of years indicates; for instance, at the end of six years we would have :

$7500 (1 X 6)6 - 7500 X 1, 26247696

110. If to the above condition of compound interest we add the condition of annual payments, we have the idea of an annuity; when these payments are supposed larger than the interest, (as in that case the whole might be reduced to simple interest,) it is evident that they must eventually consume the capital itself, and that compound interest must also be allowed upon these payments as well as upon the capital; the conditions of such contracts are therefore varied, and grounded upon various contingencies, and principally upon a combination of chances, particularly the probabilities of life, into which it cannot be our object to enter; the first principle which lies at their root is all that is intended to be shown here. The difference between the capital increased at compound interest. and the payments made, at any time, allowing the same rate of interest, is therefore the value of the annuity at that time; this will be founded upon the following investigation.

=

We shall here proceed as in the preceding section, calling the annual payment p; and supposing them to begin at the end of the first year, it will afterwards be easy to adapt the result to other conditions of payments, beginning at a later period Thus we have,