Given two terms from a geometric sequence, identify the first term and the common ratio: a5= 48 and a11= 3,072.

a5= 48 and a11= 3,072

Write out the formula for a geometric sequence:  a_n = a_1*r^(n-1), with n
                                                                                         beginning at 1.

Then, using the given info:

 a_5 = 48 is equivalent to a_48 = a_1*r^(5-1), or a_48 = a_1*r^4 = 48

a_11 = 3072 is equivalent to  3072 = a_1*r^10  =  a_1*r^4*r^6 = 3072

Solving a_1*r^4 = 48 for a_1, we get  48 / r^4.  Substitute this into the second equation to eliminate a_1:

a_1*r^4*r^6 = 3072 => (48 / r^4)*r^10 = 3072.  Then 48*r^6 = 3072, and

r^6 = 3072 / 48 = 64.     Thus, r = 6th root of 64 = 2

Now we must solve for a_1.  Recall that   48 = a_1*r^4 and subst. 2 for r:

48 = a_1*2^4  =>  48 = a_1*16.  Then a_1 = 3.

The first term is 3 and the common ratio is 2.