Answer:
[tex]a_n=281.25 \cdot 0.8^{n-1}[/tex]
or [tex]a_n=281.3 \cdot 0.8^{n-1}[/tex] (with a rounded to the nearest tenth)
Step-by-step explanation:
Geometric sequence formula: [tex]a_n=ar^{n-1}[/tex]
(where [tex]a[/tex] is the initial term and [tex]r[/tex] is the common ratio)
Given:
- [tex]a_3=180[/tex]
- [tex]a_5=115.2[/tex]
[tex]\implies a_3=ar^{2}=180[/tex]
[tex]\implies a_5=ar^{4}=115.2[/tex]
Common ratio r
[tex]\dfrac{ar^4}{ar^2}=\dfrac{115.2}{180}[/tex]
[tex]\implies r^2=0.64[/tex]
[tex]\implies r=\sqrt{0.64}[/tex]
[tex]\implies r=0.8[/tex]
Initial term a:
[tex]\implies a\cdot 0.8^{2}=180[/tex]
[tex]\implies a=281.25[/tex]
Final equation
[tex]a_n=281.25\cdot 0.8^{n-1}[/tex]
