Geometric Sequences

 

A geometric sequence is a sequence in which each term after the nonzero first term is found by multiplying the previous term by a common ratio r, where r ≠ 0 or 1.

 

Reminder: an arithmetic sequence is a sequence in which each term after the nonzero first term is found by adding a constant a to the previous term, where a > or < 0.

 

For the geometric sequence: 1, 5, 25, 125   the common ratio is 5.

 

To find the nth term of a geometric sequence, use the formula:

 

a_n = a₁ · r ^n-1 

If the sequence starts with 3, and r = 4, then a₄ = 3 · 4^4-1 = 3 · 64 = 192

 

The geometric means is a missing term between two nonconsecutive terms in a geometric sequence.

 

For example, in the sequence 1, 7, ___, 343  the 3rd term is missing.

 

use the formula: a_n = a₁ · r ^n-1   for the fourth term:

343 = 1 · r ^4-1 = r³

The cube root of 343 is 7. r = ± 7

The third term in the sequence is 7 x 7 = 49.