OF GEOMETRICAL PROGRESSION. 211. The geometrical progression is a series of terms of which each contains that which precedes it, or is contained in it, the same number of times; for example, this series3:6:12: 24: 48: 96: 192, is a geometrical progression, because each term contains that which precedes it the same number of times, which is here 2. This number of times is what we call the ratio of the progression. The four points which precede the progression have the same signification as the two points which precede the arithmetical progression. (204) But we put four of them to signify that the progression is geometrical. The progression is called increasing or decreasing, according as its terms go on augmenting or diminishing. We shall always consider the geometrical progression as increasing, because the properties are the same in both, in changing the words to multiply to those of to divide, and the words to contain to those of to be contained. Since the second term contains the first as many times as there are units in the ratio, it is composed of the first multiplied by the ratio. Since the third term contains the second as many times as there are units in the ratio, it is composed of the second multiplied by the ratio, and consequently of the first multiplied by the ratio, and again multiplied by the ratio; that is to say, of the first multiplied by the square, or second power of the ratio. Since the fourth term contains the third as many times as there are units in the ratio, it is composed of the third term multiplied by the ratio, and consequently of the first multiplied by the square of the ratio, and again multiplied by the ratio; that is to say, multiplied by the cube or third power of the ratio. For example, in the above progression, G is composed of the first term 3 multiplied by the ratio 2; 12 is composed of the first term 3, multiplied by the square 4 of the ratio 2; 24 is composed of the first term 3 multiplied by the cube 8 of the ratio 2. 212. In continuing the same reasoning, we see that any term of a geometrical progression is composed of the first multiplied by the ratio raised to a power expressed by the number of the terms which precede that term. Therefore, if the first term of the progression be a unit, each of the other terms will be formed of the ratio itself raised to a power signified by the number of the terms which precede it; for, the multiplication by the first term which is a unit, does not augment the product. * If we all the greatest term, a the least, and q the quotient or ratio, we shall have : To raise a number to a proposed power, to the seventh, for example, we must, according to the idea that we have given of powers, multiply this number by itself six successive times. Thus, to raise 2 to the seventh power, I should say, twice 2 are 4, twice 4 are 8, twice 8 are 16, twice 16 are 32, twice 32 are 64, twice 64 are 128, which is the seventh power of 2; but we can abridge the operation in divers ways: for example, I can first square 2, which gives 4 ; cube this 4, which gives 64, and multiply 64 by 2, which gives 128; or I can cube 2, which gives 8; square &, which gives 64, and multiply 64 by 2, which makes 128; in a word, it is of little importance what method we take, provided that 2 be found 7 times factor in the product. 213. The principle that we have established (212) upon the formation of any term in a geometrical progression, and the remark that we have made, may serve to calculate any term whatever of the progression, without being obliged to calculate those which precede that term. If we demand, for example, what would be the twelfth term of the progres sion 3:6:12:24: etc. As I know (212) that this twelfth term should be composed of the first, multiplied by the ratio raised to a power ex pressed by the number of the terms which precede this twelfth, I see that, to form it, I must multiply 3 by the ele venth power of the ratio 2. To form this eleventh power, I cube 2, which gives me 8; I cube 8, which gives me 512 for the ninth power, and lastly, I multiply 512, ninth power of the ratio, by 4, the second power, and I have 2048 for the eleventh power of 2; I then multiply 2048 by 3, and I have 6144 for the twelfth term of the progression. When we multiply any power of a number by any other power of that number, the sum of the indices of these two powers, is the index of the power that the number is raised to by this multiplication. Thus, 32 X33-35, that is to say, 9x27 243, which is the fifth power of 3. 214. Another application which we can make of this same principle, is to find as many geometrical means as we wish between two given numbers. If we required three geometrical means between 4 and 64, with a little attention, we should see that these three geometrical means are 8, 16, 32. In effect, 4: 8: 16: 32: 64, forms a geometrical progression; but if we proposed other numbers than 4 and 64, or if we required any other number of geometrical means, we should not find them so easily. We find them by virtue of the principle in question, as follows: The question requires to find the ratio which should reign in the progression, because when it shall be found, we shall easily form the terms in successively multiplying by this ratio. Let it be required, for example, to find nine geometrical means between 2 and 2048 2048 will then be the last term of a geometrical progression which commences with 2, and which ought to have nine terms between the first and the last: 2048 is then composed of the first term 2, multiplied by the ratio raised to a power expressed by the number of the terms which should precede 2048 hence (69) if we divide 2048 by the first term, the quotient will be the ratio raised to a power signified by the number of terms which ought to precede 2048; therefore, in seeking the root of this power, we shall have the ratio: now this power should be the tenth, for, as there must be nine terms between 2 and 2048, there are consequently ten before 2048 therefore we must extract the tenth root of the quotient found in dividing the greatest number 2048, by the least number 2. 215. As we can apply the same reasoning in every case, let us conclude in general that, to insert between two given numbers, as many geometrical means as we wish, we must divide the greatest of these two numbers by the least, which will give a quotient; we shall extract of this quotient a root signified by the number of the means augmented by a unit. Therefore, to return to our example, I divide 2048 by 2; this gives me 1024, of which I seek the tenth root,(†) which is 2. Thus, to form the means in question, I multiply the first term 2 continually by the ratio 2; and after having formed nine means, I again fall upon the number 2048, as we see here: 2:4:8: 16: 32: 64: 128: 256: 512: 1024 : 2048. In like manner, if it were required to find four geometrical means between 6 and 48, I should divide 48 by 6, and should extract the fifth root of the quotient 8; as 8 has not an exact fifth root, we can never assign exactly, in numbers, four geometrical means between 6 and 48; but we can approach to this root as near as we wish by a method analogous to that of the square and cube roots, which we shall make known in Algebra. In the mean time, it is sufficient to conceive that it is possible to find a number which, multiplied four times successsively by itself, approaches nearer and nearer to reproduce 8; and that it is the same for any other number and any other root; and hence we shall conclude that between any two numbers we can always find as many geometrical means as we would, either exactly or by an approximation carried to any degree of exactness, and this is all that is necessary to enable us to proceed to Logarithms. (†) ́We have not given a method for extracting the tenth root of a number, but it is the same with this as with the square and cube roots; the square root cannot have more than one figure when the proposed number has not more than two: the cube root ought to have but one figure when the proposed number has not more than 3; in like manner, the tenth root will never have more than one figure when the proposed number has not more than ten: it is the same with the other roots; the thirtieth, for example, will only have one figure, if the proposed number has not more than thirty figures; this is demonstrated as has been done for the square and cube roots. The sum of the terms of a geometrical progression is equal to the product of the greatest term, multiplied by the quotient, minus the least term, and the whole divided by the quotient minus 1; calling s the sum of the terms, we shall have: @xg-a Let us take, for example, the following geometrical pro gression: 26 18 54: 162(A.) which we may write thus: 2:6:: 6:18 :: i8: 54:: 54 : 162(B.) But in B. the sum of the antecedents is to the sum of the consequents as one antecedent is to its consequent. (Bezout, No. 185.) Therefore, 2+6+18+54:6+18+54+162 :: 2:6. Now it is evident that 2+6+18+54 is equal to the sum of the terms of the progression A, minus the last term 162, and that 6+18+54+162 is equal to the sum of all the terms of the first progression, minus the first; it is then evident, that if we calls the sum of all the terms of the progression A, a its first term, its last, q its quotient, we shall have: s-w: s—a :: a: aq Therefore, (sw) Xaq= (s—a) xa. Performing the multiplications indicated, and taking the value of s, we shall find: that the following table is calculated, by means of which, knowing three of the five things a, a, q, n, s, of a geometrical progression, we can always find the two others. It is by the aid of this table that we execute all the most difficult calculations of the bank and the finances, such as compound interest, annuities, etc. |