adapted from guesteb889db

1. Geometric Sequences
M4L4

2. • 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.

3. Recursive Form
• In a geometric sequence
next term = current term* r
• The mathematical way to write this is
𝑎 𝑛+1 = 𝑎 𝑛 ∙ 𝑟
Where…
• n is the number of the term
• 𝑎 𝑛 is the nth term in the sequence
• 𝑎 𝑛+1 is the (n + 1) term in the sequence
• r is the common ratio
• 𝑎1 is the starting term

4. Explicit Form
• To find any term in the geometric
sequence, use
𝒂 𝒏 = 𝒂 ∙ 𝒓 𝒏−𝟏
Where…
• n is the number of the term
• 𝒂 𝒏 is the nth term in the sequence
• 𝒂 is the first term in the sequence
• r is the common ratio

5. 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 𝒂 𝒏 = 𝒂 ∙ 𝒓 𝒏−𝟏, we will have
𝒂 𝒏 = 𝟐 ∙ (−𝟑) 𝒏−𝟏

6. 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 = -486
7th term = -486 * -3 = 1458
8th term = 1458 * -3 = -4374
Explicit way:
for the 8th term, plug in 8 for
n in our explicit equation
𝒂 𝒏 = 𝟐 ∙ (−𝟑) 𝒏−𝟏
𝒂 𝟔 = 𝟐 ∙ −𝟑 𝟖−𝟏
= 𝟐 ∙ −𝟑 𝟕
= 𝟐 ∗ −𝟐𝟏𝟖𝟕
= −𝟒𝟑𝟕𝟒

7. 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 𝒂 𝒏 = 𝒂 ∙ 𝒓 𝒏−𝟏, we will have
𝒂 𝒏 = 𝟐𝟓𝟔 ∙ ( 𝟏
𝟒) 𝒏−𝟏