That is,

The given first term plus the common difference will give the first of the required means; the first mean plus the common difference will give the second mean; and so on.

338. The formulas established in Arts. 335, 336,

are fundamental; and, since they constitute two independent equations, together containing all the five elements of an arithmetical progression, when any three of these are given, the other two may be readily determined. Thus, from these two equations we deduce other formulas; and there are in all twenty, as exhibited in the following

d+21±√(d+2 l) 2 — 8 d S

=

a+ (n − 1) d.

Hence, for the solution of questions in Arithmetical Progression, the following

RULE.

Substitute in the fundamental formulas, or such as may be deduced from them, the given quantities and reduce the result.

EXAMPLES.

1. If the first term of an arithmetical progression is 5, and the common difference 3, what is the 7th term?

Here, a 5, d= 3, and n = 7, to find 7.

=

Whence 5 + (7 − 1) 3 = 23, Ans.

=

2. Required the 4th term of an arithmetical progression, of which the first term is 10, and the common difference - 2.

Ans. 4.

3. If the first term is, the common difference 12, and the sum of the series -1, what is the number of terms?

Ans. 12.

4. The first term is 3, the number of terms 20, and the sum of the terms 440. Required the common differ

ence.

Ans. 2.

5. The sum of n terms of a series, of which the com3 no mon difference is 3, is

+2 n. What is the first term? Ans. .

6. Insert three arithmetical means between 9 and 18.

7. Required the first and last terms of a series of which the common difference is 5, the number of terms 6, and the sum 321. Ans. 41 and 66.

8. Find the sum of the odd numbers from 1 to 100. Ans. 2500.

9. A debt can be discharged in a year by paying $1 the first week, $3 the second, $5 the third, and so on ; required the last payment, and the amount of the debt. Ans. Last payment, $ 103; amount, $2704.

10. A person saves $270 the first year, $210 the second, and so on. In how many years will a person who saves every year $180 have saved as much as he? Ans. 4.

11. Two persons start together. The one travels ten leagues a day, the other eight leagues the first day, which he augments daily by half a league. After how many days, and at what distance from the point of departure, will they come together?

Ans. After 9 days, at a distance of 90 leagues.

GEOMETRICAL PROGRESSION.

339. A GEOMETRICAL PROGRESSION is a series, each term of which is equal to the preceding one, multiplied by a constant factor.

The constant factor is called the ratio of the progression.

Since the successive terms of the progression may be considered as derived from the first by continually multiplying it by the ratio, therefore, if the first term is positive, the series is increasing when the ratio is greater than 1, but the series is decreasing when the ratio is less than 1. Thus,

3, 6, 12, 24, 48,

is an increasing geometrical progression, in which the first term is 3, and the ratio 2; and

9, 3, 1, 귤,

is a decreasing geometrical progression, in which the ratio

340. In a Geometrical Progression, the LAST TERM is equal to the product of the first term by the ratio raised to a power whose exponent is one less than the number of terms.

Let a denote the first term, r the ratio, and 7 the last term; then the progression will be

That is, the nth term of the series will be

a pr-1

Also, by putting for the nth term, we have

341. To find the sum of a given number of terms in a Geometrical Progression.

Let S denote the sum of the terms; then

S = a+ar+ar2 + ar3 +. +am-2+am-1. (1)

Multiplying by r,

.....

r S = ar+ar2 + ar3 + ar2 +.....+am-1+ar. (2) Subtracting (1) from (2), and factoring,

Again, if 7 denote the last term, by Art. 340,

The sum is equal to the difference between the first term and the product of the last term by the ratio, divided by the difference between 1 and the ratio.

342. The limit to which the sum of a decreasing geometrical series approaches, as the number of terms becomes larger and larger, is called the sum of the series to infinity.

We may write the value of S, in equation (4), Art. 341, by changing the signs of the terms of the fraction, under the equivalent form,

Now, supposer less than 1; then when the number of terms, n, becomes infinitely great, or equal to ∞, the frac

tion

arn

1 r

must become infinitely small, or equal to 0, and may be rejected. Hence, when the number of terms in a decreasing geometrical series is infinite,

The sum of the terms of a decreasing geometrical series to infinity is equal to the first term divided by 1 less the ratio.

343.

To insert a given number of geometrical means between two given terms.

Let a and 7 be the two given terms, and m the number of terms to be inserted. Then we are to find m+2 terms in geometrical progression, a being the first term, and the last. Let, now, r denote the ratio; then

This determines r, and the m required means will be

The given first term multiplied by tne ratio will give the first of the required means; the first mean multiplied by the ratio will give the second mean; and so on.