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308 ALGEBRA. GEOMETRICAL PROGRESSION.

374. 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:

3, 6, 12, 24, ... 1 _ i 1 i_ a, ar, ar2, ar3, ...

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.

375. The Last, or nth Term, of a G. P. If we examine the series

a, ar, ar2, ar3, ar4, ...

we notice that in any term the index of r is always less by one than the number of the term in the series.

Thus the 3rd term is ar2; the 6th term is ar5; the 20th term is ar19; and, generally, the pth term is ar p-1

If n be the number of terms, and if l denote the last, or nth term, we have l = ar n-1

Ex. Find the 8th term of the series — i, |, — J, ...

The common ratio is ^ -4-(— •^), or — ^ ;

. •. the 8th term = - i x ( - §)' _ Iv— 2187— 129

376. Geometric Mean. When three quantities are in Geometrical Progression the middle one is called the geometric mean between the other two.