CHAPTER V. GEOMETRICAL PROGRESSION. 51. DEFINITION. Quantities are said to be in Geometrical Progression when they increase or decrease by a constant factor. Thus each of the following series forms a Geometrical Progression: The constant factor is also called the common ratio, and it is found by dividing any term by that which immediately precedes it. In the first of the above examples the common ratio is 2; in the second it is 1 3 in the third it is r. 52. If we examine the series a, ar, ar2, ar3, art, we notice that in any term the index of r is always less by one than the number of the term in the series. If n be the number of terms, and if I denote the last, or nth term, we have l=ar"-1. 53. DEFINITION. When three quantities are in Geometrical Progression the middle one is called the geometric mean between the other two. To find the geometric mean between two given quantities. Let a and b be the two quantities; G the geometric mean. Then since a, G, b are in G. P., 54. To insert a given number of geometric means between two given quantities. Let a and b be the given quantities, n the number of means. In all there will be n+2 terms; so that we have to find a series of n + 2 terms in G. P., of which a is the first and b the last. Let be the common ratio; Hence the required means are ar, ar2,... ar", where r has the value found in (1). Example. Insert 4 geometric means between 160 and 5. We have to find 6 terms in G. P. of which 160 is the first, and 5 the sixth. 55. To find the sum of a number of terms in Geometrical Progression. Let a be the first term, r the common ratio, n the number of terms, and s the sum required. Then NOTE. It will be found convenient to remember both forms given above for s, using (2) in all cases except when r is positive and greater than 1. Since arm-1=1, the formula (1) may be written = 463 From this result it appears that however many terms be taken the sum of the above series is always less than 2. Also we see that, by making n sufficiently large, we can make the fraction 1 as small as we please. Thus by taking a sufficient number of terms the sum can be made to differ by as little as we please from 2. In the next article a more general case is discussed. arr 1 -r Supposer is a proper fraction; then the greater the value of n the smaller is the value of ", and consequently of ; and therefore by making n sufficiently large, we can make the sum of In terms of the series differ from by as small a quantity as we please. a 1 -2 This result is usually stated thus: the sum of an infinite number of terms of a decreasing Geometrical Progression is i or more briefly, the sum to infinity is а 1 r a r Example 1. Find three numbers in G. P. whose sum is 19, and whose product is 216. a Denote the numbers by, a, ar; then xaxar= 216; hence a=6, and |