Question 96064
Notice if you divide 4096 by 1024 you get 4 (ie {{{4096/1024=4}}}). Notice how {{{1024/256=4}}} and {{{256/64=4}}}, etc. So each term is being divided by 4 (or multiplied by {{{1/4}}}) each time to get the next term. So that means we have a geometric sequence where {{{r=1/4}}}.


Remember a geometric sequence has the form:


{{{a[n]=a[1](r)^n}}} where {{{a[n]}}} is the nth term, {{{a[1]}}} is the first term, and r is the ratio between the terms. note: n starts at zero.


Since the first term is 4096, and the ratio is {{{1/4}}} (we previously solved for this), this means {{{a[1]=4096}}} and {{{r=1/4}}}



{{{a[n]=4096(1/4)^n}}} Plug in {{{a[1]=4096}}} and {{{r=1/4}}}



So the sequence is {{{a[n]=4096(1/4)^n}}}



You can check this answer by plugging in n=0 to get 4096


{{{a[n]=4096(1/4)^0}}}Plug in n=0 to find {{{a[n]}}} 



{{{a[n]=4096(1)}}} Evaluate {{{(1/4)^0}}}


{{{a[n]=4096}}} Simplify. Since n=0 produces {{{a[n]=4096}}} (our first number in the sequence), this partly verifies our sequence. 



And you can do this for every value of n to fully verify your answer.