« PreviousContinue »
EXAMPLES XXXVI. d.
Sum to infinity the following series : 1. 9, 6, 4,...
2. 12, 6, 3,... 1 1 1
1 1 1
4. 2' 4' ".
2 4' 8 1 2 4
6. -1, 3' 9' 27
g". 7. •9, 03, 001, ...
8. •8, – '4, -2,... Find by the method of Art. 358, the value of 9. •3. 10. •16. 11. . i. 12. •378.
13. 037. Find the series in which 14. The 10th term is 320 and the 6th term 20. 27
1 15. The 5 th term is and the gth term is 16
power of the
16. The 7th term is 625 and the 4th term -5.
9 17. The 3rd term is and the 6th term – 41.
16 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
middle term, n being the number of terms. 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. Sum the following series : 21. 32+2b, y4+4b, y® +66, ... to n terms.
3+2/2 3-212 22. 1,
to infinity 3-212'
3 1 2 23.
to 2' 3 9
25. The sum of four numbers in G.P. is equal to the common
1 ratio plus 1, and the first term is Find the numbers. 26. The difference between the first and second of four num
bers 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
HARMONICAL PROGRESSION. 359. DEFINITION. Three quantities a, b, c are said to be in Harmonical Progression when
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 Ilarmonical Progression are in Arithmetical Progression. By definition, if a, b, c are in Harmonical Progression,
.. a(b-c)=c(a - b), dividing every term by abc,
1 1 1 1
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 II.P. are generally solved by inverting the terms, and making use of the properties of the corresponding A.P.
3 Example. The 12th term of a II.P. is and the 19th term is
22 find the series.
Let a be the first term, d the common difference of the corresponding A.P.; then
5=the 12th term
=the 19th term
1 4 whence
4 5 7
3 3 1 3 and the Harmonical Progression is
4° 5° 2'7' 362. To find the harmonic mean between two given quantities. Let
a, b be the two quantities, II their harmonic mean;
1 1 1 then
are in A.P.,
1 1 1 1
2ab H =
a+6° 363. If A, G, II be the arithmetic, geometric, and harmonic means between a and b, we have proved A=
...(1). 2 G=Vab
.(2). 2ab H=
(3). a +6
a+b 2ab Therefore
2 'ati =ab
=G"?; that is, G is the geometric mean between A and H.
a + b
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,
Conversely, a series of quantities in continued proportion may be represented by X, xr, år,.......
Example 1. Find three quantities in G.P. such that their product is 343, and their sum 30.
b Example 2. If a, b, c be in H.P., prove that
btc' cta' a+b are also in H.P.
1 1 1
are in A.P.,
are in A.P.;
are in A.P.;
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 l the last; then by putting n=1, and =49 respectively, we obtain
49 / 50 (a
2 5 49
x 14 = 343.
2 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,
b+d ... each ratio
[Art. 324.] Whence
(0+ d)2=(a+c) (c+e).
1. Find the 6th term of the series 4, 2, 1},...