Question

Find the 100th AND the nth term for the following sequence. Please show work.

a. 197+7 x 3^27, 197+8 x 3^27, 197+9 x 3^27

Answer 1

nth term of this sequence is
`(197+(n+6)* 3^(27))`

and 100th term is
`(197+106* 3^(27))`.

Explanation:

The given sequence is
`(197+7* 3^(27)),(197+8* 3^(27)),(197+9* 3^(27))`

Now we will find the difference between each successive term.

Second term - First term =
`(197+8* 3^(27))-(197+7* 3^(27))`

=
`(8* 3^(27)-7* 3^(27))`

=
`3^(27)(8-7)`

=
`3^(27)`

Similarly third term - second term =
`3^(27)`

So there is a common difference of
`3^(27)`.

It is an arithmetic sequence for which the explicit formula will be

`T_(n)`=a + (n - 1)d

where
`T_(n)` = nth term of the arithmetic sequence

where a = first term of the arithmetic sequence

n = number of term

d = common difference in each successive term

Now we plug in the values to get the 100th term of the sequence.

`T_(n)=(197+7* 3^(27))+(n-1)* 3^(27)`

=
`(197+(n+6)* 3^(27))`

`T_(100)=(197+7* 3^(27))+(100-1)* 3^(27)`

=
`197+7* 3^(27)+99* 3^(27)`

=
`197+106* 3^(27)`

Therefore, nth term of this sequence is
`(197+(n+6)* 3^(27))`

and 100th term is
`(197+106* 3^(27))`.