EXERCISES.

21. In an arithmetical progression the first term is 2, the number of terms 18, and the common difference 3: required the greatest

term.

Answer, 53.

22. The least term is 6, the number of terms 21, and the common difference 3: required the greatest term.

Answer, 66.

23. If the least term be 3, the number of terms 22, and the common difference 5, what is the greatest term? Answer, 108.

24. A man bought 80 yards of Cloth: the first yard cost him 2s., and each succeeding yard 1s more, to the last: what did the last stand him in? Answer, £4. 1s.

GEOMETRICAL PROPORTION

AND PROGRESSION.

Geometrical Proportion is the ratio which quantities of the same kind have to each other, when multiplied or divided by the same quantity.

Numbers are said to be in geometrical progression when they increase or decrease by the repeated multiplication or division of the same quantity. Thus 2, 4, 8, 16, 32, &c. are an increasing geometrical progression, being repeatedly multiplied by two. And 81, 27, 9, 3, 1, are a decreasing geometrical progression, being repeatedly divided by 3.

If three numbers be in geometrical progression, the product of the extremes will be equal to the square of the

Also, if three terms be in geometrical progression, the first term will have the same proportion to the second as the second has to the third.

If four numbers be in geometrical progression, the product of the extremes will be equal to the product of the means. Thus, in 4, 8, 16, 32-(4X32)=(8×16.)

Also, if four terms be in geometrical progression, the first term will have the same proportion to the second as the third has to the fourth.

In a geometrical progression consisting of any number of terms, the product of the extremes is equal to the product of any two terms equally distant from them, or to the square of the mean term, when the number of terms is odd. Thus, in the decreasing geometrical progression, 128, 64, 32, 16, 8, 4, 2—128×2=64×4=32X8=16 squared.

PROBLEM 1st. To find one geometrical mean proportional between any two numbers.

EXERCISES.

1. Required a geometrical mean proportional between 3 and 12. Answer, 6.

2. Required a geometrical mean proportional between 25 and 81.

Answer, 45. 3. What is the geometrical mean proportional between 27 and 243 ? Answer, 81. 4. What is the geometrical mean proportional between 10 and 20? Answer, 14. 14213, &c.

PROBLEM 2d. To find two geometrical mean proportionals between any two numbers.

5. Find two geometrical means between 4 and 256.

Answer, 16 and 64.

6. Find two geometrical means between 8 and 512.

Answer, 32 and 128.

7. Find two geometrical means between 9 and 1125.

Answer, 45 and 225.

8. Find two geometrical means between 15 and 5145.

PROBLEM 3d. The first term, the ratio, and the number of terms being given, to find the last or any other term.

9. In a series of numbers in geometric progression, if the first term be 3, the ratio 2, and the number of terms 12, required the last term, Answer, 6144.

10. If I buy 12 Oranges, paying one farthing for the first, one halfpenny for the second, a penny for the third, and so on, doubling the price of each, what would the last come to? Answer, £2. 2s. 8d.

11. A Draper sold 20 yards of Cloth; the first yard for 3d., the second for 9d., the third for 27d,, &c., increasing in geometrical progression: what was the last yard sold for?

Answer, 3486784401 pence.

LOGARITHMS.

A Logarithm is the index, or exponent of a power to which a certain or invariable number must be raised, in order to equal the common natural number.

The invention of Logarithms is due to Lord Napier, Baron of Merchiston, in Scotland, who first published the Table in 1614; but Mr. Briggs, then Professor of Geometry at Gresham College, improved and reduced them to a more convenient form, which is now generally used.

The Tables are made use of to facilitate tedious arithmetical calculations.

They are founded upon a series of numbers in geometric progression, increasing in a tenfold proportion.

Thus

5.

0. 1. 2. 3. 4. Indices, or Logarithms. 1. 10. 100. 1000. 10000. 100000. Geometric Progression. The distinguishing feature of this system of Logarithms is, that the index or logarithm of 10 is 1, that of 100 is 2, that of 1000 is 3, that of 10,000 is 4, &c. Therefore the logarithm of any number between 1 and 10 will be 0 and some decimal parts, and that of a number between 10 and 100 will be 2 and some decimal parts, and so on.

Hence the integral part of any logarithm, which is usually called its index, is always less by one than the number of integers which the natural number consists of; and for decimals, it is the number denoting the distance of the first significant figure from the place of units, as seen