EXAMPLES XXXVI. d. Sum to infinity the following series: 1 4. 2 4' 8". 8 5 6. - 1, 8. 8, 4, 2,... Find by the method of Art. 358, the value of Find the series in which 14. The 10th term is 320 and the 6th term 20. 15. The 5th term is 27 and the 9th term is 1 16. The 7th term is 625 and the 4th term - 5. 9 17. The 3rd term is and the 6th term -4. 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 power of the 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. y2+2b, y++4b, y+6b,... to n terms. 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. 1 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 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, 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. Let a be the first term, d the common difference of the corresponding A.P.; then 362. To find the harmonic mean between two given quantities. Let a, b be the two quantities, II their harmonic mean; are in A.P., 1 1 1 then a' II' b 363. If A, G, HI be the arithmetic, geometric, and harmonic means between a and b, we have proved that is, G is the geometric mean between A and H. 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, xr2,....... Example 1. Find three quantities in G.P. such that their product is 343, and their sum 30. Example 2. If a, b, c be in H.P., prove that Let a be the first term, and 7 the last; then by putting n=1, and n=49 respectively, we obtain n +2, find the sum of 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, 1. Find the 6th term of the series 4, 2, 13,... |