141. In arithmetical proportion there are five numbers

to be considered :

1st. The first term, a.

2nd. The common difference, r. 3rd. The number of terms, n. 4th. The last term, 1.

5th. The sum, S.

The formulas

1=a+(n-1)r and S:
s = (a + 1 ) x n

contain five quantities, a, r, n, l, and S, and consequently give rise to the following general problem, viz: Any three of these five quantities being given, to determine the other

two.

We already know the value of S in terms of a, n, and r. From the formula

That is: The first term of an increasing arithmetical progression is equal to the last term, minus the product of the common difference by the number of terms less one.

From the same formula, we also find

ダニー

-a

That is: In any arithmetical progression, the common difference is equal to the difference between the two extremes divided by the number of terms less one.

QUEST.-141. How many numbers are considered in arithmetical proportion? What are they? In every arithmetical progression, what is the common difference equal to ?

The last term is 16, the first term 4, and the number of terms 5: what is the common difference?

2. The last term is 22, the first term 4, and the number of terms 10: what is the common difference?

Ans. 2.

142. The last principle affords a solution to the following question :

To find a number m of arithmetical means between two given numbers a and b.

To resolve this question, it is first necessary to find the common difference. Now, we may regard a as the first term of an arithmetical progression, b as the last term, and the required means as intermediate terms. The number of terms of this progression will be expressed by m+2.

Now, by substituting in the above formula, b for 1, and m+2 for n, it becomes

that is, the common difference of the required progression is obtained by dividing the difference between the given numbers a and b, by one more than the required number of

means.

QUEST.-142. How do you find any number of arithmetical means between two given numbers?

Having obtained the common difference, form the second term of the progression, or the first arithmetical mean, by The second mean

adding r, or

b-a m+1'

to the first term a.

is obtained by augmenting the first by r, &c.

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

2. Find twelve arithmetical means between 12 and 77.

143. REMARK. 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 but one and the same progression.

For, let α b.c d

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

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

'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, &c, we may conclude that all of these partial progressions form a single progression.

EXAMPLES.

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.

5. The first term is 4, the common difference 8, and the number of terms 8: what is the last term, and the sum of 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?

1

sion is 10, the common difference and the number of

3'

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. First 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 dif., or r=0,3.

11. Find the number of men contained in a triangular battalion, the first rank containing one man, the second 2, 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.