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4. The least term is 1, the greatest 2187, and the ratio 3,

required the sum of the series?

Ans, 3280.

5. The extremes are 10 and 100, and the ratio 1, in a geometrical progression; required the sum ? Ans. 550.

6. The extremes of a geometrical progression are 5 and 320, and the ratio 2; required the sum? Ans. 635.

297. Given the greatest term, number of terms, and ratio, to find the least term.

RULE. Involve the ratio to the power whose index is 1 less than the number of terms, divide the greatest term by this power, and the quotient will be the least term.

7. In a geometrical series, the greatest term is 972, the number of terms 6, and the ratio 3, required the least term? 972

1 = 5. then 315 = 243. wherefore =4, the least

Here 6 term required.

243

8. Required the least term of a geometrical progression, of which the greatest term is 1536, the ratio 2, and the number of terms 10?

Thus 10

19. then 29 512. wherefore

1536
512

= 3, the

least term.

9. In a geometrical progression the greatest term is 10.2487, the ratio 1.1, and the number of terms 5; required the least term?

Thus 5 1 = 4, and 1.1 = 1.4641, and 4

least term.

10.2487
1.4641

=7, the

10. In a geometrical progression consisting of 6 terms, the greatest term is 1024, and the ratio 4; required the least term? Ans. 1.

11. Given the greatest term 768, number of terms 9, and ratio 2, to find the least term? Ans. 3.

298. The least term, ratio, and number of terms being given, to find the greatest term.

RULE. Involve the ratio to that power whose index is one

less than the number of terms; multiply the power by the least term, and the product will be the greatest term.

12. The least term of a geometrical progression is 3, the ratio 2, and the number of terms 9; to find the greatest term? Thus 9 – 1 = 8, then Ã3 = 256, whence 256 × 3 = 768, the greatest term.

13. In a geometrical series of 5 terms, the least term is 10, and the ratio 7, required the greatest term?

Thus 5

1 = 4, and 7 = 2401, whence 2401 x 10 = 24010, the greatest term.

14. The least term of a geometrical series is 8, the ratio 3, and the number of terms 7; to find the greatest term?

Thus 36 x 8729 × 8 = 5832, the greatest term.

15. In a geometrical progression there are given the least term 2, the ratio 3, and the number of terms 4; to find the greatest term? Ans. 54.

16. Required the greatest term of a geometrical series, whose least term is 5, ratio 6, and number of terms 7? Ans. 233280.

299. The two extremes, and the sum of the series being given,

to find the ratio.

RULE. Subtract the least term from the sum, and also the greatest from the sum; then divide the former remainder by the latter, and the quotient will be the ratio required.

17. In a geometrical progression, the extremes are 10 and 10000, and the sum is 11998, required the ratio?

Thus 11998-10= 11988. and 11998 - 10000 = 1998.

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18. The extremes are 1024 and 59049, and the sum 175099; required the ratio?

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19. To find the ratio of a geometrical progression, whose sum

is 550, and the extremes 10 and 100?

540

Thus 550

10=540, and 550 — 100 = 450, whence

450

1, the ratio required.

20. The extremes are 1 and 2187, and the sum of the series

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21. The sum of a geometrical series is 635, and the extremes are 5 and 320, required the ratio?

Ans. 2.

300. The least term and ratio being given, to find any

proposed term of the series.

RULE I. Write down a few of the leading terms of the given geometrical series, and place over them as indices the terms of an increasing arithmetical series, whose common difference is 1, namely 1, 2, 3, 4, 5, &c. when the least term and ratio of the given series are equal; and 0, 1, 2, 3, 4, 5, &c. when they are unequal.

II. Add together such of the indices as will make the index of the term required; if the least term and ratio are equal, this index will be equal to the number denoting the place of that term; but if they are unequal, the index will be 1 less.

III. Multiply those terms of the geometrical series together, which stand under the indices added, and the product will be the term sought, when the first (or least) term and ratio are equal i.

IV. But if the first term be not equal to the ratio, involve the first term to the power whose index is 1 less than the number of terms multiplied, divide the above product by this power, and the quotient will be the term required.

22. The first term of a geometrical series is 2, the ratio 2, and the number of terms 14; required the last or greatest term? OPERATION. indices.

Thus 1. 2. 3. 4. 5

And 2. 4.

8. 16. 32 leading terms.

Then 2+ 3+ 4 + 2 + 3 = 14 = index of the 14th term.

This property of the indices is the foundation of Logarithms; its use in this place is extremely obvious: for knowing the last term, we also know what its index will be; and knowing the index, we readily perceive what terms of the urithmetical series must be added together to produce it, and these terms indicate what terms in the geometrical series are to be multiplied together to produce the last term.

To try to account for this mutual correspondence of the two progressions, would be a vain attempt; like many other properties, it follows from the nature of numbers, and this is perhaps all that can be said on the subject,

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And 4 x 8 x 16 x 4 x 8 = 16384 = the 14th term, or

answer.

Explanation.

The first term and ratio being equal, I take the series 1. 2. 3. 4, &c. (beginning with 1) for indices: under these I place the leading terms 2. 4. 8, &c. of the given geometrical series; then, because the index of the term required is evidently 14, I choose any of the indices, which added together make 14, namely, 2. 3. 4. 2 and 3; I then multiply the terms which stand under these together, namely, 4. 8. 16. 4 and 8, and the product is the answer required.

23. Required the 20th term of the series 1. 3. 9. 27, &c. ? OPERATION.

term.

Thus O. 1. 2. 3. 4. 5 indices.

And 1. 3. 9. 27. 81. 243 leading terms.

Then 5+ 5+ 4 + 3 + 2 = 19 index of the 20th term.
And 243 x 243 × 81 x 27 x 9 = 1162358667 the 20th

Explanation.

The first term and ratio not being equal, the indices must begin with 0, and consequently 19 will be the index of the 20th term. But by the rule, the first term ought to have been involved to the 4th power, (one less than 5, the number of terms multiplied,) by which the product of the terms should have been divided; this is omitted, because the first term being 1, all its powers will be 1, and dividing by 1 makes no alteration.

24. What is the 10th term of the series 5. 10. 20. 40, &c.?

OPERATION.

Thus O. 1. 2. 3. 4 indices.

And 5. 10. 20. 40. 80 leading terms.

Then 4 + 3 + 2 = 9 index of the 10th term.
Whence 80 x 40 x 20 = 64000 dividend.

Also 52 = 25 the divisor. wherefore

term required.

Explanation.

64000

=2560 the 10th

25

The first term 5, and ratio 2, being unequal, I divide the product of the terms, viz, 64000 by 52 or 25: that is, by that power of the first term 5, whose index 2, is less by I than the number of terms 3, multiplied together.

25. What is the 11th term of the series 1. 2. 4. 8. 16, &c.? Ans. 1024.

26. Required the 13th term of the series 2. 4. 8. 16. 32, &c. ¿ Ans. 8192.

27. The first term of a geometrical progression is 5, and the ratio 3; required the 13th term ? Ans. 2657205.

301. PROMISCUOUS EXAMPLES FOR PRACTICE.

1. Nine sea officers divide a prize, the first receives 201. the second 60l. and so on in triple proportion; what sum will the Admiral (who has the largest share) receive? Ans. 131220l.

2. Bought 12 pigs, and paid a farthing for the first, a halfpenny for the second, and so on, doubling continually the price of the last; what did they cost me? Ans. 41. 5s. 3d.4.

3. A servant agreed with his master for 12 months, to receive a farthing for the first months' service, a penny for the second, 4d. for the third, &c. what sum did his wages amount to? Answer, 58251. 8s. 5d.4.

4. The profits of a certain trading company, which has been established 12 years, have increased yearly in geometrical progression; the gain of the first year was 51. and that of the year just expired 8857351. required the ratio of increase, and the sum of the profits? Ans. the ratio 3. the sum 1328600l.

5. A person of property in Ireland agreed with Government to exert his influence, to procure seamen for the navy; the first month he sent over 1 man, the second 2 men, the third 4, and so on in geometrical progression; what number did he send over in 15 months, and how many in the last month of that time? Ans. sent in all 32767 men: in the last month 16384.

6. Suppose a laceman agrees to sell 22 yards of lace at the rate of 2 pins for the first yard, 6 for the second, and so on in triple proportion; what sum will he receive for the whole, allowing the pins to be worth a farthing a hundred? swer, 326886l. Os. Id.

An

7. What sum would a horse sell for that has 4 shoes on, with 8 nails in each shoe, at 1 farthing for the first nail, 2 for the second, 4 for the third, and so on? And what would be the price of another horse, having only two shoes, on the same conditions? Ans. 44739241. 5s. 3d. the first: and 681. 5s. 3d. the last.

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