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GEOMETRICAL PROGRESSION. 309 377. 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., b G = G a

each being equal to the common ratio; G2 = ab; whence G = ab.

378. To insert a given number of geometric means between two given quantities.

Let a and b be the given quantities, m the number of means.

There will be m + 2 terms; so that we have to find a series of m + 2 terms in G. P., of which a is the first and b the last.

Let r be the common ratio;

then b = the (m+2)th term = ar m + 1; r m + 1 = b a r = (b a) 1 m +1 (1).

Hence the required means are ar, ar2,..., arm, where r has the value found in (1).

Ex. 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.

Let r be the common ratio; then 5 = the sixth term = 160 r5 ; r5 = 1 32 > whence, by trials r = 1 2;

and the means are 80, 40, 20, 10.