Question

Question #9a098

Answer 1

Thus at `n=7" we have "a_7=3xx4^7 = 49152`

Explanation:

They will be looking for you to derive an equation that gives the value of any term

Let the term count be `i`
Let the ith term be `a_i`
Let the last term be `a_n`

`color(brown)("Test for arithmetic sequence:")`
`12-3=9`
`48-12 = 26`

`9!=12` so this is not an arithmetic sequence. Thus is could be a geometric one.

`color(brown)("Test for geometric sequence:")`

`12-:3=4`
`48-:12=4`
`192-:48=4`

The values are all the same so this is a geometric sequence that involves 4 raise to some power

Try:

`i=1->a_1=3xx4^1=12`
`i=2->a_2=3xx4^2=3xx16= 48`
`i=3->a_3=3xx4^3=3xx64=192`

This works so we have: for any `ncolor(white)(" ")a_n=3xx4^n`

Thus at `n=7" we have "a_7=3xx4^7 = 49152`