Answer:
Substitute n =1, 2, 3, 4, 5 to get the first five terms.
Step-by-step explanation:
(4). [tex]$ g_n = 2 . 3^{n - 1} $[/tex]
To find the first term, substitute n = 1.
Therefore, [tex]$ g_1 = 2 . 3^{1 - 1} = 2 . 3^{0} = 2 . 1 $[/tex] = 2
[tex]$g_2 = 2 . 3^{2 - 1} = 2 . 3 $[/tex] = 6
[tex]$ g_3 = 2 . 3^{3 - 1} = 2 . 3^{2} = 2 . 9 $[/tex] = 18
[tex]$ g_4 = 2 . 3^{4 - 1} = 2 . 3^{3} = 2 . 27 $[/tex] = 54
[tex]$ g_5 = 2 . 3^{5 - 1} = 2 . 3^{4} = 2 . 81 $[/tex] = 168
Therefore the first five terms of the sequence are: 2, 6, 18, 54, 168.
(5). [tex]$ t_n = \frac{2}{3}t_{n - 1} $[/tex]
[tex]$ t_2 = \frac{2}{3} t_{2 - 1} = \frac{2}{3}t_1 = \frac{2}{3}6 $[/tex] = 4
[tex]$ t_3 = \frac{2}{3} t_2 = \frac{2}{3}. 4 = \frac{8}{3} $[/tex]
[tex]$ t_4 = \frac{2}{3}t_3 = \frac{2}{3}\frac{8}{3} = \frac{16}{9} $[/tex]
[tex]$ t_5 = \frac{2}{3}t_4 = \frac{2}{3}\frac{16}{9} = \frac{32}{27} $[/tex]
Therefore the first five terms in the sequence are: 6, 4, 8/3, 16/9, 32/27.
