A geometric function can be represented explicitly or recursively.
The recursive rule is: [tex]\mathbf{a_n = -4a_{n-1} }[/tex]
The sequence is given as:
[tex]\mathbf{4, -16, 64, -256.....}[/tex]
The terms are represented as:
[tex]\mathbf{a_1 = 4}[/tex]
[tex]\mathbf{a_2 = -16}[/tex]
[tex]\mathbf{a_3 = 64}[/tex]
[tex]\mathbf{a_4 = -256}[/tex]
Rewrite as:
[tex]\mathbf{a_1 = 4}[/tex]
[tex]\mathbf{a_2 = 4 \times -4}[/tex]
[tex]\mathbf{a_3 = -16 \times -4}[/tex]
[tex]\mathbf{a_4 = 64 \times -4}[/tex]
Substitute [tex]\mathbf{a_3 = 64}[/tex] in [tex]\mathbf{a_4 = 64 \times -4}[/tex]
[tex]\mathbf{a_4 = a_3 \times -4}[/tex]
Express 3 as 4 - 1
[tex]\mathbf{a_4 = a_{4-1} \times -4}[/tex]
Substitute n for 4
[tex]\mathbf{a_n = a_{n-1} \times -4}[/tex]
Evaluate the product
[tex]\mathbf{a_n = -4a_{n-1} }[/tex]
Hence, the recursive rule is: [tex]\mathbf{a_n = -4a_{n-1} }[/tex]
Read more about geometric functions at:
https://brainly.com/question/222209
