The second term of a geometric sequence is -18 and the fifth term is 2/3. What is the sixth term?
2 Answers
Explanation:
#"using the nth term formula for a geometric sequence"#
#•color(white)(x)a_n=ar^(n-1)#
#rArra_2=ar=-18to(1)#
#rArra_5=ar^4=2/3to(2)#
#"divide equation "(2)" by equation "(1)#
#rArr(ar^4)/(ar)=(2/3)/(-18)=-1/27#
#rArrr^3=-1/27rArrr=root(3)(-1/27)=-1/3#
#"substitute this value in equation "(1)#
#axx-1/3=-18rArra=54#
#rArra_6=54xx(-1/3)^5=54xx-1/243=-2/9#
Explanation:
The
#a_n = a r^(n-1)#
where
In our example, we find:
#-1/27 = (2/3)/(-18) = a_5/a_2 = (ar^4)/(ar) = r^3#
The only real solution of this is:
#r = root(3)(-1/27) = -1/3#
If we permit complex common ratios then there are two other possibilities, namely
Assuming we want to use the real solution:
#a_6 = a_5 * r = -1/3(2/3) = -2/9#