Geometric Sequences

M4L4

• Geometric sequence – a sequence in which terms are found by multiplying a preceding term by a nonzero constant.

{ 3, 9, 27, ___,…} r = 3

• The constant r is called the common ratio.

Recursive Form

• In a geometric sequence next term = current term* r

• The mathematical way to write this is

Where…

• n is the number of the term• is the nth term in the sequence• is the (n + 1) term in the sequence• r is the common ratio• is the starting term

Explicit Form

• To find any term in the geometric sequence, use

Where…• n is the number of the term• is the nth term in the sequence• is the first term in the sequence• r is the common ratio

Example 1

• Find the explicit formula for the geometric sequence

2, -6, 18, -54,…

Our starting term a = 2, and our common ratio is r = -3.

To put it in explicit form , we will have

Example 2

• Given the geometric sequence 2, -6, 18, -54,… find the 8th term.

Recursive way: the 4th term is -54 so 5th term = -54 * -3 = 162 6th term = 162 * -3 = -4867th term = -486 * -3 = 14588th term = 1458 * -3 = -4374

Explicit way: for the 8th term, plug in 8 for n in our explicit equation

Example 3

• Find the explicit formula for the geometric sequence

256, 64, 16, 4, 1…

Our starting term a = 256, and our common ratio is r = ¼ .

To put it in explicit form , we will have