Respuesta :
Answer:
nth term of this sequence is [tex](197+(n+6)\times 3^{27})[/tex]
and 100th term is [tex](197+106\times 3^{27})[/tex].
Step-by-step explanation:
The given sequence is [tex](197+7\times 3^{27}),(197+8\times 3^{27}),(197+9\times 3^{27})[/tex]
Now we will find the difference between each successive term.
Second term - First term = [tex](197+8\times 3^{27})-(197+7\times 3^{27})[/tex]
= [tex](8\times 3^{27}-7\times 3^{27})[/tex]
= [tex]3^{27}(8-7)[/tex]
= [tex]3^{27}[/tex]
Similarly third term - second term = [tex]3^{27}[/tex]
So there is a common difference of [tex]3^{27}[/tex].
It is an arithmetic sequence for which the explicit formula will be
[tex]T_{n}[/tex]=a + (n - 1)d
where [tex]T_{n}[/tex] = 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.
[tex]T_{n}=(197+7\times 3^{27})+(n-1)\times 3^{27}[/tex]
= [tex](197+(n+6)\times 3^{27})[/tex]
[tex]T_{100}=(197+7\times 3^{27})+(100-1)\times 3^{27}[/tex]
= [tex]197+7\times 3^{27}+99\times 3^{27}[/tex]
= [tex]197+106\times 3^{27}[/tex]
Therefore, nth term of this sequence is [tex](197+(n+6)\times 3^{27})[/tex]
and 100th term is [tex](197+106\times 3^{27})[/tex].
