Answer:
r = 2
Step-by-step explanation:
nth term of a geometric sequence = [tex] ar^{n - 1} [/tex].
Where,
a = first term, r = common ratio,
We are given that, the 11th term is 32 times larger than the 6th term of the geometric sequence. Thus:
the 6th term can be expressed as [tex] ar^{n - 1} = ar^{6 - 1} [/tex]
[tex] a_6 = ar^{5} [/tex]
The 11th term is expressed as [tex] a_11 = ar^{10} [/tex]
Since the 11th term is 32 times larger than the 6th term, therefore, the following equation can be derived to find the common ratio, r, of the sequence:
[tex] a_11 = 32*a_6 [/tex]
[tex] a_11 = ar^{10} [/tex]
[tex] a_6 = ar^{5} [/tex], therefore:
[tex] ar^{10} = 32*ar^{5} [/tex]
Divide both sides by ar⁵
[tex] \frac{ar^{10}}{ar^5} = \frac{32*ar^{5}}{ar^5} [/tex]
[tex] r^5 = 32 [/tex]
[tex] r^5 = 2^5 [/tex]
[tex] r = 2 [/tex]
Common ratio = 2
