Hence ram-nẞn-m, and therefore a = am−n ßn-m ̧ Ex. Find the first term of the G.P. whose 3rd term is 18 and whose 5th term is 403. If a be the first term, and r the common ratio, we have 223. When three quantities are in G. P., the middle one is called the Geometric Mean of the other two. If a, b, c are in G. P., we have by definition. Thus the geometric mean of two given quantities is a square root of their product. When any number of quantities are in geometrical progression all the intermediate terms may be called geometric means of the two extreme terms. Between two given quantities any number of geometric means may be inserted. For let a and b be the two given quantities, and let n be the number of means to be inserted. Then b will be the (n + 2)th term of a G. P. of which a is the first term. Hence, if r be the common ratio, we have Hence the required means are ar, ar2,................ ar”, 224. To find the sum of any number of terms in geometrical progression. Let a be the first term, and r the common ratio. Let n be the number of the terms whose sum is required, and let / be the last of them. Then, since is the nth term, we have l= ar”-1. Ex. 1. Find the sum of 10 terms of the series 3, 6, 12, &c. Now when r is a proper fraction, whether positive or negative, the absolute value of r" will decrease as n increases; moreover the value of r" can be made as small as we please by sufficiently increasing the value of n. Hence, when r is numerically less than unity, the sum of the series can be made to differ from 1 α by as small a quantity as we please by taking a sufficient number of terms. S. A. 17 Thus the sum of an infinite number of terms of the geometrical progression a + ar+ar2+......, in which r is numerically less than unity, is α Ex. 1. Find the sum of an infinite number of terms of the series 9-6+4..... Ex. 2. Find the geometrical progression whose sum to infinity is 4, and whose second term is -2. Let a be the first term, and r be the common ratio. is inadmissible, for r must be numerically less than 3 unity. Ex. 3. The 3rd term of a G.P. is 2, and the 6th term is ; what is the 10th term? Ans. -4. Ex. 4. Insert two geometric means between and 1, and three means between 2 and 18. Ans. -4, 2; ±2√/3, 6, ±6\/3. Ex. 5. Shew that if all the terms of a G.P. be multiplied by the same quantity, the products will be in G. P. Ex. 6. Shew that the reciprocals of the terms of a G.P. are also in G.P. Ex. 7. Shew that, if between every two consecutive terms of a G.P., a fixed number of geometric means be inserted, the whole will form a geometrical progression. Ex. 9. Shew that the continued product of any number of quantities n in geometrical progression is equal to (gl)2, where n is the number of the quantities and g, l are the greatest and least of them. Ex. 10. Shew that the product of any odd number of terms of a G.P. will be equal to the nth power of the middle term, n being the number of the terms. Ex. 11. The sum of the first 10 terms of a certain G.P. is equal to 244 times the sum of the first 5 terms. What is the common ratio? Ans. 3. Ex. 12. If the common ratio of a G.P. be less than, shew that each term will be greater than the sum of all that follow it. 226. HARMONICAL PROGRESSION. Definition. A series of quantities is said to be in Harmonical Progression when the difference between the first and the second of any three consecutive terms is to the difference between the second and the third as the first is to the third. Thus a, b, c, d &c., are in Harmonical Progression [H. P.], if If a, b, c be in harmonical progression, we have by definition Hence, dividing by abc, we have 1 which shews that 1 1 b' c b are in arithmetical progression. Thus, if quantities are in harmonical progression, their reciprocals are in arithmetical progression. 227. Harmonic Mean. If a, b, c be in harmonical progression, Hence a' b' Thus the harmonic mean of two quantities is twice their product divided by their sum. If we put A, G, H for the arithmetic, the geometric, and the harmonic means respectively of any two quantities a and b, we have Thus the geometric mean of any two quantities is also the geometric mean of their arithmetic and harmonic means. 228. Theorem. The arithmetic mean of two unequal positive quantities is greater than their geometric mean. If a, b be the two positive quantities we have to shew that |