Answer: Given that a, b, and c form an arithmetic sequence, so they are equally spaced, meaning the difference between each of them is constant. Let's denote the common difference between a, b, and c as d.
So, we have:
a = b - d
c = b + d
Using these two equations, we can prove that 2^a, 2^b, and 2^c form a geometric sequence.
We have:
2^a = 2^b / 2^d
2^c = 2^b * 2^d
Since the product of two powers of the same base is equivalent to the power of the base with the sum of the exponents, we have:
2^a * 2^c = 2^b * 2^b * 2^d * 2^d = 2^(b+b+d+d) = 2^(2b+2d)
So, we have:
2^a * 2^c = 2^(2b+2d) = 2^(2b) * 2^(2d) = 2^b * 2^b * 2^d * 2^d = 2^b * 2^c
This means that 2^a, 2^b, and 2^c form a geometric sequence, with the common ratio of 2^d.
Explanation: