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(c.) If the series is formed by addition, it is called an INCREASING Series; if by subtraction, it is called a DECREASING SERIES.

(d.) The numbers composing a series are called the Terms of the series.

(e.) The difference between the consecutive terms of any series is called the Common DIFFERENCE, and is always the number by the addition or subtraction of which the series is formed.

(f.) Since the terms of a series are formed by continual additions or subtractions of the same number, it follows that the second term of any series equals the first, plus or minus the common difference; that the third term equals the first, plus or minus twice the common difference; that the fourth term equals the first, plus or minus three times the common difference, etc.

(g.) Hence, any term of an arithmetical series is equal to the first term, plus or minus the common difference taken one less times than there are terms in the series ending with the required term.

(h.) Moreover, if the first term of an increasing arithmetical series be subtracted from the last, or if the last term of a decreasing series be subtracted from the first, the remainder will be the product of the common difference multiplied by one less than the number of terms. .

1. What is the 12th term of the increasing series of which 1 is the first term and 8 is the common difference? (See g.)

2. What is the 6th term of the decreasing series of which 40 is the first term and 4 is the common difference ?

3. What is the common difference of the series of which 1 is the first term and 25 the 8th? (See h.)

4. What is the common difference of the series of which 50 is the first term and 24 is the 14th ?

5. How many terms are there in the series having 3 for the first term, 31 for the last, and 2 for the common difference?

6. How many terms are there in the series having 56 for the first term, 32 for the last, and 3 for the common difference ?

7. Form a series of 4 terms, of which 3 is the first term and 9 is the last.

8. Form a series of 9 terms, of which 10 is the first and 6 is the last.

145. To find the Sum of an Arithmetical Series.

(a.) If we should invert any series, and add the first term of the result to the first term of the original series, the second term of the result to the second term of the original series, etc. each successive sum would equal the sum of the first and last term of the series. But all these sums taken together would equal the sum of both series, or (since the series are equal twice the sum of either series. ILLUSTRATION. — By inverting the series 5, 8, 11, etc. to 20, we have —

5 8 11 14 17 20 = given series.
20 17 14 11 8 5 = same series inverted.

25 25 25 25 25 25 = sums of the successive terms. Hence, the sum of both series equals 6 X 25, and the sum of one of them equals 1 of 6 X 25, which is 3 X 25 = 75.

(b.) Hence, the sum of a series in arithmetical progression equals half the product obtained by multiplying the sum of the first and last terms by the number of terms.

1. What is the sum of a series of 12 terms, of which 2 is the first term and 35 the last ?

2. What is the sum of a series of 10 terms, of which 81 is the first term and 36 is the last ?

3. What is the last term and the sum of a series of 15 terms,

4. What is the last term and what the sum of a series of 16 terms, of which 90 is the first term and 4 is the common difference ?

6. What is the sum of the numbers from 1 to 100 ?

6. What is the number of terms and what the common difference of the series of which 42 is the sum, 2 is the first term, and 12 is the last ?

7. What is the number of terms and what the common difference of a series of which the sum is 121, the first term is 16, and the last term 6?

8. What is the last term and what the common difference of a series of 12 terms, of which the sum is 138, and the first term is 67

9. What is the first term and what the common difference of a series of 10 terms, of which the last term is 10, and the sum is 85 ?

146. Geometrical Progression.

(a.) A series of numbers in geometrical progression, or a geometrical series, is a series of numbers, each of which bears the same ratio to the one which follows it.

(b.) Such a series would be obtained by continually multiplying by the same number.

IllustratIONS.—Beginning with 3 and multiplying by 2, we should have the series 3, 6, 12, 24, 48, 96, etc.

Beginning with 1728 and multiplying by , we have the series 1728, 432, 108, 27, etc.

(c.) The numbers comprising the series are called the TERMS OF THE SERIES.

(d.) The ratio of each term to that which follows it is called the Common Ratio, and is always the number by which we should multiply to produce the series.

(e.) In an increasing series, the common ratio equals a whole number or an improper fraction; but in a decreasing series, the common ratio equals a proper fraction.

(f.) From the method of forming such a series, it is obvious that, the second term must equal the first multiplied by the common ratio ; that the third term must equal the first multiplied by the second power of the common ratio, etc.

(g.) Hence, any term of a geometrical series must equal the product of the first term multiplied by the common ratio raised to a power one degree less than the number of that term.

(h.) Moreover, if the last term of a geometrical series be divided by the first, the quotient will be the common difference raised to a power one degree less than the number of the term.

1. What is the 9th term of the series of which 1 is the first term and 2 the common ratio ?

2. What is the 8th term of the series of which al is the first term and 3 the common ratio ?

3. What is the 10th term of the series of which 81 is the first term and j the common ratio ?

4. What is the common ratio of a series of which 2 is the first term and 512 the 5th ?

5. What is the common ratio of a series of which 15625 is the first term and 4096 the 7th ?

6. Construct the series of which 16 is the first term and 10000 the 5th.

147. To find the Sum of a Geometrical Series.

(a.) If a geometrical series should be multiplied throughout by the common ratio, all the terms except the last of the series thus derived would correspond to all the terms except the first of the given series. Hence, the difference between the original and the derived series would equal the difference between the first term of the original series and the last term of the derived series. But, from the manner in which the derived series is found, this difference must also equal the product of the original series multiplied by the difference between 1 and the common ratio. Hence, dividing the difference between the first term and the common ratio times the last term, by the difference between the common ratio and 1, must give the sum of the series.

ILLUSTRATIONS.-Multiplying the series 3, 12, etc. to 768 by 4, we have

a = 3, 12, 48, 192, 768 = given series.

4 a = 12, 48, 192, 768, 3072 = derived series. 4a - a= 3 a = 3072 - 3 = 3069 = 3 times given series. Hence, 3069 + 3 = 1023 = sum of the series.

Multiplying the series 18, 6, etc. to by }, we have

a = 18, 6, 2, 4, 2, 2 = given series.

a = 6, 2, , , , ,i = derived series. a-fa = a = 18 – = of the given series. Hence, 1731 + f = 26.= sum of the series. (b.) It thus appears that, to find the sum of a geometrical series, we may multiply the last term by the common ratio, and divide the difference between the product and the first term by the difference between 1 and the common ratio.

1. What is the sum of a series of which 1 is the first term, 256 is the last, and 2 is the common ratio ?

2. What is the sum of a series of which 486 is the first term, & is the last, and } is the common ratio ?

8. What is the common ratio and what the sum of a series of 4 terms, of which 9 is the first term and 64827 is the last ?

4. What is the common ratio and what the sum of a series of 4 terms, of which 3267 is the first term and 121 the last ?

5. If the last term is 64, the number of terms is 8, and the common ratio is ļ, what is the sum of the series ?

6. If the first term of a geometrical series is 2, the number of terms is 11, and the common ratio is 2, what is the last term, and what the sum of the series ?

148. Infinite Decreasing Series.

(a.) Since each term of a decreasing series is smaller than the preceding, it follows that if the series be carried far enough, the terms will become so small that they may be disregarded without affecting sensibly the sum of the series.

(b.) An infinite decreasing series will always be of this character, and hence its sum will equal the quotient obtained by dividing the first term by the difference between 1 and the common ratio.

1. What is the sum of the infinite decreasing series of which 6 is the first term and the common ratio ?

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