Geometric Sequences

What is a Geometric Sequence?

A sequence is geometric when the ratios of consecutive terms are the same.
Tip

The common ratio is always the quotient between two consecutive terms.
Is this a geometric sequence?

1. 6, 12, 24, 48, …

Answer

Yes, r = 2.

2. 1, 8, 27, 64

Answer

No.

3. $\displaystyle{\frac{3}{5}}, \displaystyle{\frac{4}{25}}, \displaystyle{\frac{5}{125}}, \displaystyle{\frac{6}{625}}$ , ….

Answer

No.

4. $2,\displaystyle{\frac{4}{3}}, \displaystyle{\frac{8}{9}}, \displaystyle{\frac{16}{27}},$ …

Answer

Yes, $r = \displaystyle{\frac{2}{3}}$

5. $3, 1, \displaystyle{\frac{1}{3}}, \displaystyle{\frac{1}{9}}, \displaystyle{\frac{1}{27}},$ …

Answer

Yes, $r = \displaystyle{\frac{1}{3}}$
Find the missing term in the geometric sequence

1. 2, ____, 32, 128,

Answer

8, $r = 4$

2. -4, _____, ______, 32, -64, 128, ____

Answer

8, -16, -256 $r = -2$

3. $3x^2,3x^4,3x^6,3x^8$ , ____

Answer

$3x^{10},r=x^2$

4. $3x, 6x^2, 12x^3, 24x^4,$ ____

Answer

$48x^5,r=2x$