Answer:
[tex] a_n = 16(\frac{1}{4})^{n - 1} [/tex]
Step-by-step explanation:
Given:
Fifth term of a geometric sequence = [tex] \frac{1}{16} [/tex]
Common ratio (r) = ¼
Required:
Formula for the nth term of the geometric sequence
Solution:
Step 1: find the first term of the sequence
Formula for nth term of a geometric sequence = [tex] ar^{n - 1} [/tex], where:
a = first term
r = common ratio = ¼
Thus, we are given the 5th term to be ¹/16, so n here = 5.
Input all these values into the formula to find a, the first term.
[tex] \frac{1}{16} = a*\frac{1}{4}^{5 - 1} [/tex]
[tex] \frac{1}{16} = a*\frac{1}{4}^{4} [/tex]
[tex] \frac{1}{16} = a*\frac{1}{256} [/tex]
[tex] \frac{1}{16} = \frac{a}{256} [/tex]
Cross multiply
[tex] 1*256 = a*16 [/tex]
Divide both sides by 16
[tex] \frac{256}{16} = \frac{16a}{16} [/tex]
[tex] 16 = a [/tex]
[tex] a = 16 [/tex]
Step 2: input the value of a and r to find the nth term formula of the sequence
nth term = [tex] ar^{n - 1} [/tex]
nth term = [tex] 16*\frac{1}{4}^{n - 1} [/tex]
[tex] a_n = 16(\frac{1}{4})^{n - 1} [/tex]
