Question

Question #07e0c

Answer 1

Assumption: you mean the tenth term

`a_10->1/27xx3^(10-1) = 729`

Explanation:

First of all observe that `1/27<1/9<1/3 < ? " "`and so on.
So as we progress through the sequence the values are increasing

Notice the following:

`3xx3=9`
`3xx9=27`

So to progress from one term to the next the value of 3 in some format is involved.

Also notice that:

`1/27xx3=1/9`

`1/27xx3xx3=1/3`

Set the position in the sequence be designated as the `n^("th")` position.
Let the `n^("th")` term be designated as `a_n`

When `n=1` we are looking at the first place `a_1->1/27`
When `n=2` we are looking at the second place `a_2->1/9`

and so on. So putting this all together we have:

Note that `3^0=1=4^0=5^0` and so on

`a_n=a_1 ->1/27xx1 color(white)(---)->color(white)(---)1/27xx3^0`
`a_n=a_2->1/27xx3`
`a_n=a_3->1/27xx3xx3`

So by inspection we observe that for any `n>0` we have
`a_n=1/27xx3^(n-1)`

So `a_10->1/27xx3^(10-1) = 729`