To solve this question, we need to understand the concept of compound interest and its formula.


The formula to calculate the compound interest is given by:

A = P(1 + r/n)^(n*t)

Where:
A = Final amount
P = Principal amount (initial sum)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years

In this question, it is given that the sum becomes k times in 6 years at compound interest. So, let's assume the initial sum (principal amount) as P. After 6 years, the final amount will be kP.


Now, we need to find the interest rate (r) from this information. We can rearrange the compound interest formula to solve for r:

r = ([(A/P)^(1/(n*t))]) - 1

Substituting the values, we have:
r = [(kP/P)^(1/(n*t))] - 1
r = k^(1/(n*t)) - 1


Now, we need to calculate how many times the initial sum will become in 24 years. Let's call this factor X.

Using the compound interest formula, we have:
X = (1 + r/n)^(n*t)

Substituting the values, we have:
X = (1 + k^(1/(n*t)) - 1/n)^(n*t)

Simplifying the expression, we have:
X = (k^(1/(n*t)))^n

Since n*t = 6 (as given in the question), we can substitute this value:
X = (k^(1/6))^n

Now, let's simplify further:
X = (k^(1/6))^n
X = k^(n/6)

Since n = 24 (as given in the question), we can substitute this value:
X = k^(24/6)
X = k^4

Therefore, after 24 years, the sum will become k^4 times the initial amount. Hence, the correct answer is option 'B' - k^4.