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816 ALGEBRA. 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. 365.]

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. 374.]

4. If a, b, c, d ... be in G. P., they are also in continued proportion, since by definition

a b = b c = c d = ... = 1 r

Conversely, a series of quantities in continued proportion may be represented by x, xr, xr2, ... .

Ex. 1. Find three quantities in G. P. such that their product is 343, their sum 30 13.

Let a r, a, ar be the three quantities; then we have

a r a  ar = 343 (1),

and

a ( 1 r + 1 + r) = 91 3 (2). From (1) a3 = 343 a = 7;

hence from (2) 7(1 + r + r2) = 91 3 r. Whence we obtain r = 3, or 1 3 ; and the numbers are 7 3, 7, 21.

Ex. 2. If a, b, c be in H. P., prove that a b+c, b c+a, c a+b are also in H.P.

Since 1 a, 1b, 1c are in A. P. , a+b+c a, a+b+c b, a+b+c c are in A. P. ; 1 + b+c a, 1 + c+a b, 1 + a+b c are in A.P. b+c a, c+a b , a+b c are in A. P. ; a b+c, b c+a, c a+b are in H. P.