GEOMETRICAL PROGRESSION.

A GEOMETRICAL PROGRESSION is a series of numbers, the terms of which gradually increase or decrease by the constant multiplication or division of some particular number; as 1, 3, 9, 27, 81, 243, &c., or 64, 32, 16, 8, 4, 2, 1, 1, &c.

In the first case, the series is increasing by the constant multiplication of 3; in the second, it is a decreasing series. by the constant division of 2. It is evident that both series may be carried on for ever.

The number by which the series is constantly increased or diminished is called the ratio.

Any three of these terms being given or known, the others may be determined.

I. Given the first term a, the last term z, and the common ratio r, to find the sum s.

RULE. Multiply the last term by the ratio, and from the product subtract the first term, and the remainder divided by the ratio, less one, will give the sum of the series; or

Ex. 1. The first term of a series in geometrical progression is 5, the last term is 3645, and the ratio is 3: what is the sum?

For the terms are 5, 15, 45, 135, 435, 1215, and 3645; which,

being added together, make 5465.

Ex. 2. The first and last terms of a geometrical series are 4 and 3294179 and the common ratio is 7: what is the sum?

Ex. 3. The first and last terms of a geometrical progression are 4 and 262144, and the ratio 4: what is the sum?

GEOMETRICAL PROGRESSION.

151

II. Given the first term a, the number of terms n, and the ratio r, to find the last term z.

The last term may be obtained by continual multiplication; but as that, in a long series, is a tedious process, we shall give the following rule:)

1. When the first or least term is equal to ratio.

RULE. Write down some of the leading terms of the geometrical series, over which place the arithmetical series 1, 2, 3, 4, &c., as indices;* find what figures of these indices added together will give the index of the term wanted in the geometrical series; then multiply the numbers, standing under such indices, into each other, and their product will be the term sought.

Ex. 1. What is the last term of a geometrical series having 13 terms, of which the first is 2, and the ratio 2?

Here the series, with their indices, will stand thus:

2', 4, 8, 16, 325, 64°, &c.

The number of terms being 13, the index to the last term will be 13 equal to the indices+ 5+ 6, which figures standing over 4, 32, and 64, shew that these last are to be multiplied together, and the product is the term sought; thus 4 X 32 X 648192.

Ex. 2. What is the last term of the series having 9 terms, of which the first is 3, and the ratio 3?

Ex. 3. What did the last of 12 oxen cost, the first of which was sold for 3s., the second for 9s., and so on?

2. When the first term a, of the series, is not equal to the ratio r.

RULE. Write down the leading terms of the series, and place their indices over them, beginning with a cypher, add together the most convenient indices to make an index less one than the number expressing the place of the term sought; then multiply the numbers standing under such

NOTE.

*When the natural numbers 1, 2, 3, 4, 5, &c., are set over a geometrical series, they are called indices, or exponents, and they shew the distance of any term from unity, or from the first term: thus, in the series 2', 4o, 83, 16a, 643, 128°, &c., 1, 2, 3, &c. are the indices, and shew the distance of any term of the series from the first term; the index 5, for instance, shews that 64 is the fifth term in the series.

indices, into each other, dividing the product of every two by the first term in the geometrical series; the last quotient is the term required.

Ex. 1. What is the last term of the series, whose first term is 4, ratio 3, and number of terms 15?

4°, 12', 36°, 108, 324*, 972%, 2916, &c.

The number of terms being 15, the index sought must be 14 equal to 653, under which stand the terms 2916, 972, and 108, then 2916 X 972 708588 X 108

708588, and

19131876 z last

4

term.

Ex. 2. The first term of a geometrical series is 2, the number of terms 12, and the ratio 5, required the last term?

Ex. 3. The first term of a geometrical series is 1, the ratio 2, and the number of terms 25, what is the last term, and also the sum of all the terms?

Ex. 4. The first term of a series is 5, the ratio 3, and the number of terms 16, what is the last term, and the sum of the terms?

Ex. 5. A hosier sold 12 pair of stockings, the first pair at 3d., the second 9d., and so on in geometrical progression; for what did he sell the last pair, and how much had he for the whole?

Ex. 6. What would a horse fetch, supposing it was sold on condition of receiving for it one farthing for the first nail in his shoes, a halfpenny for the second, one penny for the third, and so on, doubling the price of every nail to 32, the number in his four shoes?

Ex. 7. A husbandman agreed to serve his master during hay-time and harvest, or five-and-forty clear days, provided he would give him a barley-corn only for the first day's work, 3 for the second, o for the third, and so on in geometrical proportion; what would he have to receive in money for his labours, supposing there were half a million of grains in a bushel, and each bushel was worth 4s. ?

The following facts may be committed to memory:

1. If three numbers are in geometrical progression, the product of the extremes is equal to the square of the means; as 3, 9, 27, here 3 × 279 × 981.

2. If four numbers are in geometrical progression, the product of the extremes is equal to the product of the means; as 2, 4, 8, 16; here 2 × 76 = 4 × 8 = 32.

3. If the series contain an odd number of terms, the square of the middle term is equal to the product of the adjoining extremes, or of any two terms equally distant from them; as 3, 9, 27, 81, 243; here 2723 × 243 = 9 × 81.

LOGARITHMS.

LOGARITHMS are artificial numbers, invented for the purpose of facilitating certain tedious arithmetical operations.

If any series of numbers in arithmetical progression beginning with 0, be taken, and a corresponding series of geometrical numbers beginning with 1, the former series will be logarithms to the corresponding numbers in the latter; thus,

0, 1, 2, 3, 4,

5, 6, 7, 8, 9 logarithms.

1, 2, 4, 8, 16, 32, 61, 128, 256, 512 numbers.

Here 0, 1, 2, &c. are the logarithms of 1, 2, 4, &c., and it will be seen at once, 1. That Addition in logarithms answers to Multiplication in common numbers :

Thus, if the logarithms 2 and 6 are added together, the sum is 8 which answers to the logarithm of 256, the number that is obtained by the multiplication of 4 and 64, which are the numbers standing under the logarithms 2 and 6. By adding the logarithms 4 and 5 we have 9, which stands over 512, the number obtained by multiplying together 16 and 32. Hence the addition of logarithms answers to multiplication in common numbers.

3

2. Subtraction in logarithms answers to division of common numbers.

Divide 256 by 8, and you have 32, over which stands 5 = 8 the logarithms standing above.

3;

3. Multiplication in logarithms answers to involution of common numbers.

Ex. The square of 8 is 64; now 3 is the logarithm answering to 8, and 3 × 2, (because 2 is the index of the square,) is equal to 6, which is the logarithm of 64.

4. Division in logarithms answers to evolution in common arithmetic.

Ex. 1. The square root of 256 is 16, over which stands the logarithm 4; which answers to 82, 8 being the logarithm of 256.

Ex. 2. The cube root of 512 is 8; and 9, which is the logarithm of 512, divided by 3, the sign of the cube, gives 3, which is the. logarithm of 8.

The same indices will serve for any geometric series; but the

logarithms generally made use of are those which increase in a tenfold proportion, as

0. 1.

2.

3.

4.

5.

6. &c. 1. 10. 100. 1000. 10000. 100000. 1000000.

Here it is evident, that the logarithms of numbers between 1 and 10, are greater than 0, and less than one, as will be seen in the table at the end of the volume, thus the logarithms of 2, 6, 8, &c. are, .3010300, .7781513, .9030900, &c.

The logarithms of the numbers between 10 and 100, are greater than 1, and less than 2; thus the logarithm of 15 is 1.1760913, and. the logarithm of 95 is 1.9777236.

The logarithms of numbers between 100 and 1000, are greater than 2, and less than 3; thus the logarithm of 165 is 2.2174839, and of 984 is 2.9920951.

The logarithms between 1000 and 10000, must be somewhere between 3 and 4; between 10.000 and 100.000 they must be between 4 and 5; and so on.

The logarithms in the above series are called indices, which are frequently neglected, the decimal part only being put down; thus, if it be required to find the logarithm of 248, it will be sufficient to put down .3944517, and the number being between 100 and 1000, I know the index is 2. Therefore the rule for finding the index is his :

The index is always one less than the number of figures in the whole number; or the figures in the whole number must be always one more than the index.

Here the decimal figures remain the same; and the only difference is in the indices, which are increased or diminished by unit for every ten-fold increase or decrease of the whole number. It will be observed, that where there is but one whole number, the index will be 0;" but if the figures be decimals, as .248, the index is minus one, or— 1; if by the prefixing o to the decimal figure, their value is diminished in a tén-fold proportion, then the index is -2, or minus two: it there are two cyphers on the left of the decimal, then the index is-3, minus three, and so on.

We shall now proceed to shew the use of Logarithms and the manner of working by them; and give some instances of their application.