1) The general form of for the #nth# of a geometric sequence with first term #a_1#, common ratio between all terms #r# is given by
#a_n=a_1(r)^(n-1)#
Given #a_1=2, r=4,# we plug these values into the above formula, yielding a general form of the geometric sequence which we can use to get any term(s) we want:
#a_n=2(4)^(n-1)#
We know the first term #a_1=2.# That has been given.
For the second term, plug in #n=2# into #a_n=2(4)^(n-1):#
#a_2=2(4)^(2-1)#
#=2(4)^1=2(4)#
#=8#
Repeat this process for the third, fourth, and fifth terms:
#a_3=2(4)^(3-1)#
#=2(4)^2#
#=2(16)#
#=32#
#a_4=2(4)^(4-1)#
#=2(4)^3#
#=2(64)#
#=128#
#a_5=2(4)^(5-1)#
#=2(4)^4#
#=2(256)#
#=512#
Thus, the first #5# terms are #2, 8, 32, 128, 512.#
2) Again, plug in #a_1=48, r=-1/3# into #a_n=a_1(r)^(n-1)#, which gives us
#a_n=48(-1/3)^(n-1)#
We know the first term #a_1=48.# To determine the second, third, fourth, and fifth terms, simply plug in #n=2, n=3, n=4, n=5# into the above formula:
#a_2=48(-1/3)^(2-1)#
#=48(-1/3)=-16#
#a_3=48(-1/3)^(3-1)#
#=48(-1/3)^2#
#=48/9=16/3#
#a_4=48(-1/3)^(4-1)#
#=48(-1/3)^3#
#=-48/27=-16/9#
#a_5=48(-1/3)^(5-1)#
#=48(-1/3)^4#
#=48/81=16/27#
Thus, the first #5# terms are #48, -16, 16/3, -16/9, 16/27.#
