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5.

EXAMPLES XXXVI. d.

15. The 5th term is

2. 12, 6, 3,...

1

1 1

2

4' 8"

1

2

4

3' 9' 27'

7. 9, 03, 001,...

8.

Find by the method of Art. 358, the value of

9. ·3. 10. 16.

11. i 12. 378.

4.

23.

6.

Find the series in which

14. The 10th term is 320 and the 6th term 20.

8

5

5

-1, ğ"**

8, 4, 2,...

27
16

16. The 7th term is 625 and the 4th term - 5.
9

17. The 3rd term is

and the 6th term -41.

16

3+2√2 3-2/2

3-2√2'

3+2/2

2

9

1

and the 9th term is

18. Divide 183 into three parts in G.P. such that the sum of the first and third is 2 times the second.

19. Show that the product of any odd number of consecutive terms of a G.P. will be equal to the nth power of the middle term, n being the number of terms.

Sum the following series:

21. y2+2b, y+4b, yo+6b,... to n terms.

22.

1,

20. The first two terms of an infinite G.P. are together equal to 1, and every term is twice the sum of all the terms which follow. Find the series.

to infinity.

1

√ √ √3... to infinity.

2' 3

13. 037.

24. 2n-1, 4n+1, Cn-1,... to 2n terms.

18

1

25. The sum of four numbers in G.P. is equal to the common ratio plus 1, and the first term is Find the numbers. 17 26. The difference between the first and second of four numbers in G.P. is 96, and the difference between the third and fourth is 6. Find the numbers.

27. The sum of $225 was divided among four persons in such a manner that the shares were in G.P., and the difference between the greatest and least was to the difference between the means as 21 to 6. Find the share of each. 28. The sum of three numbers in G.P. is 13, and the sum of 13 Find the numbers. 9

their reciprocals is

HARMONICAL PROGRESSION.

359. DEFINITION. Three quantities a, b, c are said to be in

α a-b

Harmonical Progression when

с b-c'

Any number of quantities are said to be in Harmonical Progression when every three consecutive terms are in Harmonical Progression.

360. The reciprocals of quantities in Harmonical Progression are in Arithmetical Progression.

By definition, if a, b, c are in Harmonical Progression,

dividing every term by abc,

1

с

=

α α

с

-C

.. a(b-c)=c(a - b),

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;

1

which proves the proposition.

361. Harmonical properties are chiefly interesting because of their importance in Geometry and in the Theory of Sound: in Algebra the proposition just proved is the only one of any importance. There is no general formula for the sum of any number of quantities in Harmonical Progression. Questions in H.P. are generally solved by inverting the terms, and making use of the properties of the corresponding A.P.

Example. The 12th term of a II.P. is

find the series.

Let a be the first term, d the common difference of the corresponding A.P.; then

and

whence

5 the 12th term

=a+11d;

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Therefore

=a+18d;

1

d=3

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Hence the Arithmetical Progression is

are in A.P.,

a =

4 5
3'3

3 1 3

and the Harmonical Progression is 4' 5' 2' 7'

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362. To find the harmonic mean between two given quantities. Let a, b be the two quantities, II their harmonic mean;

then

1 1 1
a' II' ī

=

AII=

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1

b II'

2 =ab

=

2ab

a+b

a+b

2

G=√ab

2ab

H=

a+b

a+b 2ab

a+b

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363. If A, G, I be the arithmetic, geometric, and harmonic means between a and b, we have proved

A=

;

= C2;

that is, a is the geometric mean between A and H.

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...(1).

..(2).

.(3).

364. Miscellaneous questions in the Progressions afford scope for much skill and ingenuity, the solution being often very neatly effected by some special artifice. The student will find the following hints useful.

1. If the same quantity be added to, or subtracted from, all the terms of an A.P., the resulting terms will form an A.P. with the same common difference as before. [Art. 342.]

2. If all the terms of an A.P. be multiplied or divided by the same quantity, the resulting terms form an A.P., but with a new common difference. [Art. 342.]

3. If all the terms of a G.P. be multiplied or divided by the same quantity, the resulting terms form a G.P. with the same common ratio as before. [Art. 352.]

4. If a, b, c, d... be in G.P., they are also in continued proportion, since, by definition,

then we have

and

Conversely, a series of quantities in continued proportion may be represented by x, xr, xr2,.................

,"

From (1)

Example 1. Find three quantities in G.P. such that their product is 343, and their sum 30}.

Let, a, ar be the three quantities;

xaxar=313

... from (2)

a Ն с

C

d

Whence we obtain

=

a

91

a ( + 1+r) = ? 31

and the numbers are , 7, 21.

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(1),

...(2).

Example 2. If a, b, c be in H.P., prove that

are also in H. P.

1 1 1

Since

a' b'c

are in A.P.,

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.. each ratio

Whence

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[ocr errors]

b+c

a

b+c

a

[ocr errors]
[ocr errors]

1+

a + c

b

a + c

b

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1

a=5+2,

=

1+

a

b

C

b+c' c+a' a+b

a+b

с

8 = 12 (a + 1):

49

a+b

C

are in H.P.

n

Example 3. The nth term of an A.P. is +2, find the sum of

49 terms.

Let a be the first term, and the last; then by putting n=1, and n=49 respectively, we obtain

49 1=

are in A.P.;

5

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are in A.P.;

49/50

2 5

b

с

b+c'c+a' a+b

+2;

d

are in A.P.;

=2×14=343.

Example 4. If a, b, c, d, e be in G.P. prove that b+d is the geometric mean between a + c and c+e.

Since a, b, c, d, e are in continued proportion,

a

b с
t с d e

=

b+d

a + c
=

'b+d c+e'
(b+d)2=(a+c) (c+e).

;

[Art. 324.]

EXAMPLES XXXVI. e.

1. Find the 6th term of the series 4, 2, 13,...

2. Find the 21st term of the series 2, 113, 11%,...

9

3. Find the 8th term of the series 13, 11, 2,... 4. Find the nth term of the series 3, 1, 1, ...

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