Question

what is the recursive formula for a geometric sequence is an=2a n-1 with an initial value of an=2a n-1 with an initial value of a1=1/8 what is the explicit formula for the sequence?

Answer 1

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: