Question

What is the explicit formula for this geometric sequence?

f 1, 2, 3, 4, 5,
f(n) 1, 4, 8, 16, 32

Answer 1

The formula for any geometric sequence is

$$b_n=b_1q^{n-1},$$

where $b_1$ - the first element, $q$ - common ratio, $n$ - number of an element.

Here we have sequence numbers from 1 to 5, i.e. 5 elements, and 5 values for these sequence elements.

Looking at the equation above, we can notice that according to the condition either there should be 6 elements in f row and as well as we must include 2 into f(n), or it must be 2 instead of 1 in f(n) row. There is no any geometric sequence with the numbers you wrote.

Therefore, assuming case a):

$$n\space\space\space\space\space1 \space2\space 3\space 4\space \space5\space \space\space6 \\ f(n)\space1 \space2 \space4 \space8 \space16 \space32\\ b_n=1\cdot2^{n-1}=2^{n-1}.$$

Assuming case b):

$$n\space\space\space\space\space\space1 \space2\space 3\space\space 4\space \space5\space\\ f(n)\space\space2 \space4 \space8 \space16 \space32\\ b_n=2\cdot2^{n-1}=2^n.$$