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

Accepted Solution

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)dwhere [tex]T_{n}[/tex] = nth term of the arithmetic sequencewhere a = first term of the arithmetic sequencen = number of termd = common difference in each successive termNow 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].