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2. Find the sum of 12 terms of the series 6, 9, 12, etc.

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Find the sum

3. Of 16 terms of the series 2, 4, 6, etc. 4. Of 20 terms of the series 1, 3, 5, etc. 5. Of 30 terms of the series 2, 6, 10, etc. 6. Of 40 terms of the series 1, 6, 11, etc. 7. Of 24 terms of the series 2, 8, 14, etc. 8. Of 100 terms of the series 1, 4, 7, etc.

9. How many times does a common clock strike in a week? 10. A traveled 15 miles the first day, 18 miles the second, 21 miles the third, and so on for 36 days; how far did he travel ?

11. A boy received $5 for January, $10 for February, $15 for March, and so on; how much did he receive for the year?

12. If a body falls 161 ft. the first second, 3 times as far the second second, 5 times as far the third second, and so on, how far will it fall in 25 seconds ?

13. 100 apples are placed in a row 2 yd. apart, the first being 2 yd. from a basket: a boy starts to gather them singly into the basket; how far does he travel ?

GEOMETRICAL PROGRESSION.

270. A Geometrical Progression is a series of numbers which increase or decrease by a common Ratio.

Thus, 1, 3, 9, 27, etc. is an increasing geometrical progression, in which the ratio is 3.

And 32, 16, 8, 4, etc. is a decreasing geometrical progression, in which the ratio is 1.

The numbers composing the series are called the Terms.

271. To find the last term of a geometrical progression.

WRITTEN EXERCISES.

1. Find the 7th term of the series 2, 6, 18, etc. PROCESS.

The first term is 2. 1st 2

The second term equals 2 times the 2d 2 x 3 6

first power of the ratio.
3d 2 x 32 18
4th

The third term equals 2 times the 2 x 33 54

second power of the ratio. Hence 7th 2 x 36 1458 Hence the seventh term equals 2

times the sixth power of the ratio, which equals 2 x 36 1458.

From this problem we derive the

RULE.

The last term equals the first term multiplied by the ratio raised to a power denoted by the number of terms less one. 2. Find the 6th term of the series 3, 9, 27, etc.

PROCESS.—Last term 3 x 36 2187, Ans. 3. Find the 10th term of the series 1, 2, 4, etc. 4. Find the 8th term of the series 5, 15, 45, etc. 5. Find the 7th term of the series 3, 12, 48, etc. 6. Find the 10th term of the series 1, 1, 4, etc. 7. Find the 10th term of the series 1, ș, , etc. 8. Find the 9th term of the series }, }, }, etc.

9. A man agreed to labor at the rate of $1 for January, $2 for February, $4 for March, and so on, and to accept for his year's services as much as would be due him under this agreement for December; how much would he receive?

10. A boy agreed to put one cent in his savings bank on New Year's day, and on each succeeding day of the month to double the previous day's deposit; if he had carried out his agreement, what would have been the last day's deposit ?

272. To find the sum of a geometrical progression.

WRITTEN EXERCISES. 1. Find the sum of 5 terms of the series 2, 6, 18, etc.

PROCESS.
Sum 2 + 6 + 18 + 54 + 162
3 sum

6 + 18 + 54 + 162 + 486
(3 1) sum 486 - 2

486 2 (162 x 3) 2 Sum

242, Ans. 3 1

3 - 1

Multiplying the sum of the series by 3, the ratio, and subtracting the series, we have (3 – 1) times the sum of the series equal to 486 – 2, and the sum of the series equal (486 – 2) divided by (3 – 1).

Observing that 162 is the last term, 3 the ratio, and 2 the first term, we have the

RULE.

The sum of a geometrical progression equals the last term multiplied by the ratio, minus the first term, divided by the ratio less

опе. .

2. Find the sum of 8 terms of the series 3, 6, 12, etc.

PROCESS.—Last term 3 x 27 384.

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3. Of 1, 2, 4, etc. to 10 terms.
4. Of 2, 8, 32, etc. to 9 terms.
5. Of 6, 12, 24, etc. to 12 terms.
6. Of 5, 15, 45, etc. to 14 terms.
7. Of 10, 20, 40, etc. to 10 terms.

8. If the population of a city is doubled every 5 years, what is its population at the end of 50 years, if it begins with 5000 inhabitants ?

9. A agreed to labor 21 days on condition that he should receive 1 cent for the first day, 2 cents for the second, 4 cents for the third, and so on; how much did he receive in all ?

10. A gentleman, thinking $5000 too much for a farm containing 17 acres, agreed to pay 1 cent for the first acre, 3 cents for the second, 9 cents for the third, and so on; which price was the greater, and how much?

INFINITE SERIES.

273. An Infinite Series is a series which has an infinite number of terms.

In a decreasing geometrical progression we subtract the last term multiplied by the ratio from the first term, and divide by 1 minus the ratio.

In a decreasing geometrical progression of an infinite number of terms the last term becomes so small that it may be regarded as zero; hence from the above we have the following

RULE.

The sum of an infinite series equals the first term divided by 1 minus the ratio.

WRITTEN EXERCISES.

1. Find the sum of the infinite series 1 + i + 4, etc.

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Find the sum

2. Of the infinite series 1 + i + ġ, etc.
3. Of the infinite series 1 + 1 + 1b, etc.
4. Of the infinite series 1 + io + Too, etc.
5. Of the infinite series + i + ', etc.

SECTION XIII.

MISCELLANEOUS PROBLEMS. 274.-1. Find the greatest common divisor of 1107, 1353, and 1599.

2. Find the least common multiple of 369, 697, and 861.
3. Reduce 59} to its lowest terms.
4. Reduce 42 + 3 to an improper fraction.

.8
5. Reduce .6 of 4 of to a simple fraction.

of 6
6. Add + 4 + 2 to 12 - 18

180 7. Find the value of of of of divided by á of of 3. 8. What part of 44 is 27?

23 9. What part of

is .41? 31

4. 26 10. Divide 16; by

of

9 63 11. Add 37 + 187 + 42% + 100%.

X

153.

of

3} - 1 12 13. Divide .00084 by 57000. 14. Divide 7000 by .00002;. 15. Reduce it and fi to decimals.

110 16. Find the value of (.25 = 500 + 7.5) X

.0053

12. Reduce 3} x 3} – 1

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