Answer:
Arithmetic Sequence
[tex]T_n = 240-60n[/tex] ---- Explicit
[tex]T_n = T_{n-1} - 60[/tex] --- Recursive
Geometric Sequence
[tex]T_n = 270* (\frac{2}{3})^n[/tex] ---- Explicit
[tex]T_n = T_{n-1} * \frac{2}{3}[/tex] ---- Recursive
Step-by-step explanation:
Given
[tex]180, 120,....[/tex]
(a) Assume it is an arithmetic sequence
The explicit formula is calculated using:
[tex]T_n = a + (n - 1)d[/tex]
Where
[tex]a = 180[/tex]
[tex]d = 120 - 180[/tex]
[tex]d = -60[/tex]
So, we have:
[tex]T_n = 180 + (n - 1)*-60[/tex]
[tex]T_n = 180 -60n + 60[/tex]
Rewrite
[tex]T_n = 180 + 60-60n[/tex]
[tex]T_n = 240-60n[/tex]
The recursive function is calculated using:
[tex]T_1 = 180[/tex]
[tex]T_2 = 120 = 180 - 60 = T_1 - 60[/tex]
[tex]T_3 = 60 = 120 - 60 = T_2 -60[/tex]
-
[tex]T_n = T_{n-1} - 60[/tex]
(b) Assume it is a geometric sequence
The explicit formula is calculated using:
[tex]T_n = ar^{n-1}[/tex]
Where
[tex]a = 180[/tex]
[tex]r = \frac{120}{180}[/tex]
[tex]r = \frac{2}{3}[/tex]
So, we have:
[tex]T_n = 180 * (\frac{2}{3})^{n-1}[/tex]
Split
[tex]T_n = 180 * (\frac{2}{3})^n \div (\frac{2}{3})^1[/tex]
[tex]T_n = 180 * (\frac{2}{3})^n \div (\frac{2}{3})[/tex]
Rewrite as:
[tex]T_n = 180 * (\frac{2}{3})^n * (\frac{3}{2})[/tex]
[tex]T_n = 180 * (\frac{3}{2})* (\frac{2}{3})^n[/tex]
[tex]T_n = 180 * \frac{3}{2}* (\frac{2}{3})^n[/tex]
[tex]T_n = 90 * 3* (\frac{2}{3})^n[/tex]
[tex]T_n = 270* (\frac{2}{3})^n[/tex]
The recursive function is calculated using:
[tex]T_1 = 180[/tex]
[tex]T_2 = 120 = 180 * \frac{2}{3} = T_1 * \frac{2}{3}[/tex]
[tex]T_3 = 80 = 120 * \frac{2}{3} = T_2 * \frac{2}{3}[/tex]
-
[tex]T_n = T_{n-1} * \frac{2}{3}[/tex]
