To solve this problem, we need to understand what an exponential sequence is and how it behaves.
An exponential sequence is a sequence in which each term is obtained by multiplying the previous term by a constant ratio. In other words, each term can be expressed as a = a1 * r^(n-1), where 'a' is the term, 'a1' is the first term, 'r' is the common ratio, and 'n' is the term number.
In this case, we are given that 3, p, q, and 24 are consecutive terms of an exponential sequence. Let's assign the first term, 3, to 'a1'. Since the terms are consecutive, we can assign 'n' as follows: 3 is the first term (n = 1), p is the second term (n = 2), q is the third term (n = 3), and 24 is the fourth term (n = 4).
So, for each term, we have:
a1 = 3
a2 = 3 * r^(2-1) = 3 * r
a3 = 3 * r^(3-1) = 3 * r^2
a4 = 3 * r^(4-1) = 3 * r^3
We also know that the consecutive terms are given as 3, p, q, and 24.
Therefore, we can write the equations as follows:
3 * r = p
3 * r^2 = q
3 * r^3 = 24
To find the values of p and q, we need to solve these equations.
From the first equation, we can express 'r' in terms of 'p': r = p / 3.
Substituting this in the second equation, we have: 3 * (p / 3)^2 = q
Simplifying, we get: p^2 = q
From the third equation, we have: 3 * (p / 3)^3 = 24
Simplifying, we get: p^3 = 24
Taking the square root of both sides of p^2 = q, we get: p = sqrt(q).
Substituting this value of p in the equation p^3 = 24, we have: (sqrt(q))^3 = 24
Simplifying, we get: sqrt(q^3) = 24
Taking the cube of both sides, we have: q^3 = 24^2 = 576
Taking the cube root of both sides, we get: q = 6
Substituting the value of q in the equation p = sqrt(q), we have: p = sqrt(6)
Therefore, the values of p and q are approximately p ≈ sqrt(6) and q ≈ 6.