Question

Sydney placed 10 guppies in a fish tank she recorded the number of guppies in the tank

Answer 1

`g(n) = 10 * 3^(n)`

Explanation:

Given

See attachment for table

Required

Determine the explicit formula for the table

First, we need to check if the guppies increases at arithmetic progression or geometry progression.

For arithmetic progression:

We calculate the common difference (d)

`d = g(n) - g(n-1)`

Take n as 1

`d = g(1) - g(1-1)`

`d = g(1) - g(0)`

From the table:

`g(1) = 30\ \&\ g(0) = 10`

`d = 30 - 10`

`d = 20`

Take n as 2

`d = g(2) - g(1)`

From the table:

`g(1) = 30\ \&\ g(2) = 90`

`d = 90 - 30`

`d = 60`

The two differences are not equal. So, this

For geometry progression:

We calculate the common ratio (r)

`r = (g(n))/(g(n-1))`

Take n as 1

`r = (g(1))/(g(1-1))`

`r = (g(1))/(g(0))`

From the table:

`g(1) = 30\ \&\ g(0) = 10`

`r = (30)/(10)`

`r = 3`

Take n as 2

`r = (g(2))/(g(1))`

From the table:

`g(1) = 30\ \&\ g(2) = 90`

`r = (90)/(30)`

`r = 3`

The two ratios are equal.

So, the next step is to calculate the explicit formula using n term of a geometric progression.

Because the progression starts from 0, we make use of the following formula

`g(n) = g(0) * r^(n)`

Substitute values for g(0) and r

`g(n) = 10 * 3^(n)`

Hence:

The explicit formula is:
`g(n) = 10 * 3^(n)`