Now, by substituting in the above formula, b for l, and m+2 for n, it becomes

The common difference of the required progression is obtained by dividing the difference between the given numbers, a and b, by the required number of means plus one.

Having obtained the common difference, d, form the second term of the progression, or the first arithmetical mean, by adding d to the first term a. The second mean is obtained by augmenting the first mean by d, etc.

(1) Find three arithmetical means between the extremes 2 and 18.

(2) Find twelve arithmetical means between 12 and 77. ·

184. If the same number of arithmetical means are inserted between all the terms, taken two and two, these terms, and the arithmetical means united, will form one and the same progression.

Let a.b.c.e.f..... be the proposed progression, and m the number of means to be inserted between a and b, b and c, c and e....., etc.

From what has just been said, the common difference of each partial progression will be expressed by

expressions which are equal to each other, since a, b, c, ...... are in progression. Therefore the common difference is the same in each of the partial progressions; and, since the last term of the first forms the first term of the second, etc., we may conclude that all of these partial progressions form a single progression.

Exercises.

1. Find the sum of the first fifty terms of the progression 2.9.16.23 .....

2. Find the 100th term of the series 2.9.16.23.....

Ans. 695.

3. Find the sum of 100 terms of the series 1.3.5.7.9

Ans. 10000.

4. The greatest term is 70, the common difference 3, and the number of terms 21. What is the least term, and the sum of the series? Ans. Least term, 10; sum of series, 840. is 4, the common difference 8, and the What is the last term, and the sum of Ans. Last term, 60; sum of series, 256.

5. The first term number of terms 8. the series?

6. The first term is 2, the last term 20, and the number of terms 10. What is the common difference?

Ans. 2.

7. Insert four means between the two numbers 4 and 19. What is the series?

Ans. 4.7. 10. 13. 16. 19.

8. The first term of a decreasing arithmetical progression is 10, the common difference, and the number of terms 21. Required the sum of the series. Ans. 140.

9. In a progression by differences, having given the common difference 6, the last term 185, and the sum of the terms 2945, find the first term and the number of terms.

Ans. 1st term, 5; number of terms, 31.

10. Find nine arithmetical means between each antecedent and consequent of the progression 2.5.8.11. 14.....

Ans. Common difference, or d, 0.3.

11. Find the number of men contained in a triangular battalion, the first rank containing 1 man, the second 2 men, the third 3, and so on to the nth, which contains n: in other words, find the expression for the sum of the natural numbers 1, 2, 3 ....., from 1 to n inclusively. n(n+1). 2

Ans. S=

12. Find the sum of the n first terms of the progression of uneven numbers 1.3.5.7.9 Ans. Sn2.

13. One hundred stones being placed on the ground in a straight line at the distance of 2 yards apart, how far will a person travel who shall bring them one by one to a basket placed at a distance of 2 yards from the first stone?

Ans. 11 miles, 840 yards.

GEOMETRICAL PROPORTION AND PROGRESSION.

185. Ratio is the quotient arising from dividing one quantity by another quantity of the same kind, regarded as a standard. Thus, if the numbers 3 and 6 have the same unit, the ratio of 3 to 6 will be expressed by

6

and in general, if A and B represent quantities of the same kind, the ratio of A to B will be expressed by

B

A

186. The character x indicates that one quantity varies as another.

Thus,

A ∞ B

is read, “A varies as B;" that is to say, the ratio of A to B, B or is constant, though A and B themselves may change Α'

value.

If there are four numbers, 2, 4, 8, 16, having such values that the second divided by the first is equal to the fourth divided by the third, the numbers are said to form a proportion; and in general, if there are four quantities, A, B, C, and D, having such values that

then A is said to have the same ratio to B that C has to D, or the ratio of A to B is equal to the ratio of C to D. When four quantities have this relation to each other, compared together two and two, they are said to form a geometrical proportion.

To express that the ratio of A to B is equal to the ratio of C to D, we write the quantities thus:

A: B:: C: D,

and read, "A is to B as C to D."

The quantities which are compared the one with the other are called terms of the proportion. The first and the last terms are called the two extremes; and the second and the third terms, the two means. Thus, A and D are the extremes; and B and C the means.

187. Of four terms of a proportion, the first and the third are called the antecedents; and the second and the fourth, the consequents; and the last is said to be a fourth proportional to the other three taken in order. Thus, in the last proportion, A and C are the antecedents, and B and D the consequents.

188. Three quantities are in proportion when the first has the same ratio to the second that the second has to the third; and then the middle term is said to be a mean proportional between the other quantities. For example,

3:6:6:12;

and 6 is a mean proportional between 3 and 12.

189. Four quantities are said to be in proportion by inversion, or inversely, when the consequents are made the antecedents, and the antecedents the consequents.

Thus, if we have the proportion

3:68:16,

the inverse proportion would be

6:3:16:8.

190. Quantities are said to be in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent.

Thus, if we have the proportion

3:68:16,

the alternate proportion would be

3:8:6:16.