Answer:
Li Jing's formula i.e. [tex]\boxed{g_n=-5\cdot \:5^{n-1}}[/tex] is right.
Step-by-step explanation:
Considering the sequence
[tex]-5,\:-25,\:-125,\:-625,...[/tex]
A geometric sequence has a constant ratio r and is defined by
[tex]g_n=g_0\cdot r^{n-1}[/tex]
[tex]\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{g_{n+1}}{g_n}[/tex]
[tex]\frac{-25}{-5}=5,\:\quad \frac{-125}{-25}=5,\:\quad \frac{-625}{-125}=5[/tex]
[tex]\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}[/tex]
[tex]r=5[/tex]
So, the sequence is geometric.
as
[tex]\mathrm{The\:first\:element\:of\:the\:sequence\:is}[/tex]
[tex]g_1=-5[/tex]
[tex]r=5[/tex]
so
[tex]g_n=g_1\cdot r^{n-1}[/tex]
[tex]\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:[/tex]
[tex]g_n=-5\cdot \:5^{n-1}[/tex]
Therefore, Li Jing's formula i.e. [tex]\boxed{g_n=-5\cdot \:5^{n-1}}[/tex] is right.
