Question

Mohamed and Li Jing were asked to find an explicit formula for the sequence -5, -25, -125, -625,....

Mohamed said the formula is g(n) = -5.5", and
Li Jing said the formula is g(n) = -5.50 -1.
Which one of them is right?

Answer 1

Li Jing's formula i.e.
`\boxed{g_n=-5\cdot \:5^(n-1)}` is right.

Explanation:

Considering the sequence

`-5,\:-25,\:-125,\:-625,...`

A geometric sequence has a constant ratio r and is defined by

`g_n=g_0\cdot r^(n-1)`

`\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=(g_(n+1))/(g_n)`

`(-25)/(-5)=5,\:\quad (-125)/(-25)=5,\:\quad (-625)/(-125)=5`

`\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}`

`r=5`

So, the sequence is geometric.

as

`\mathrm{The\:first\:element\:of\:the\:sequence\:is}`

`g_1=-5`

`r=5`

so

`g_n=g_1\cdot r^(n-1)`

`\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:`

`g_n=-5\cdot \:5^(n-1)`

Therefore, Li Jing's formula i.e.
`\boxed{g_n=-5\cdot \:5^(n-1)}` is right.