Answer: To find the explicit formula for the geometric sequence with the recursive formula an = 2an-1 and a1 = 1/8, we can use the following steps:
First, we can write out the first few terms of the sequence using the recursive formula:
a1 = 1/8
a2 = 2a1 = 2/8 = 1/4
a3 = 2a2 = 2/4 = 1/2
a4 = 2a3 = 2/2 = 1
a5 = 2a4 = 2
We can see that the common ratio between any two consecutive terms is 2, so we can write:
a2/a1 = 2
a3/a2 = 2
a4/a3 = 2
a5/a4 = 2
In general, we can write the ratio of any two consecutive terms as:
an/an-1 = 2
We can use this to find an expression for an in terms of a1:
a2/a1 = 2
a2 = 2a1
a3/a2 = 2
a3 = 2a2 = 2(2a1) = 4a1
a4/a3 = 2
a4 = 2a3 = 2(4a1) = 8a1
a5/a4 = 2
a5 = 2a4 = 2(8a1) = 16a1
In general, we can write:
an = 2^(n-1) * a1
Substituting a1 = 1/8, we get:
an = 2^(n-1) * (1/8) = 1/(2^(8-n))
Therefore, the explicit formula for the geometric sequence is:
an = 1/(2^(8-n))
Explanation: