Geometric Sequence Recursive Formula Worksheet

A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous one by a fixed, non-zero constant called the common ratio. In this worksheet, we will explore how to find the recursive formula for a geometric sequence.

Let's start with an example. Suppose we have a sequence that begins with 168 and each subsequent term is found by multiplying the previous one by one half. How can we define this sequence explicitly in terms of N?

One way to think about it is to view the exponents as the number of times we multiply by one half. For instance, the second term is obtained by multiplying the first term (168) by one half once, resulting in 84. To get from 84 to 42, we multiply by one half again.

We can represent this sequence using a recursive formula, which defines each term in terms of the previous one. In this case, our function G(N) defines the sequence where N is the term number, starting with G(1) = 168.

Here's a table illustrating how we can apply the recursive formula to find the corresponding values for different N:

| N | G(N) |

| --- | --- |

| 1 | 168 |

| 2 | ? |

| 3 | ? |

| 4 | ? |

To find the value of G(2), we multiply the first term (168) by one half, resulting in 84. For G(3), we multiply G(2) by one half, getting 42. Similarly, for G(4), we multiply G(3) by one half, obtaining 21.

By continuing this process, we can generate more terms of the sequence:

| N | G(N) |

| --- | --- |

| 1 | 168 |

| 2 | 84 |

| 3 | 42 |

| 4 | 21 |

| ... | ... |

The recursive formula for this geometric sequence is thus: G(N) = G(N-1) × 0.5, where G(1) = 168.

In conclusion, the recursive formula allows us to define a geometric sequence by repeatedly applying a fixed operation (multiplying by one half in this case) to each term, starting with an initial value. This formula provides a powerful tool for generating and analyzing sequences with a wide range of applications in mathematics, science, and other fields.

Geometric sequence recursive formula worksheet