Let , P( 1 + r )^5 = 2P
( 1 + r )^5 = 2

now cube both the sides,
P( 1+ r )^15 = 8P

Hence, n = 15 years.

Given:
- The sum is compounded annually.
- The sum doubles in 5 years.

To find:
- In how many years will the sum become 8 times?

Solution:

Step 1: Understand the problem
- We are given that the sum doubles in 5 years.
- This means that the sum grows by a factor of 2 in 5 years.

Step 2: Determine the growth rate
- To find the growth rate, we can use the formula for compound interest: A = P(1 + r/n)^(nt), where:
- A is the final amount
- P is the principal amount (the initial sum)
- r is the annual interest rate (unknown)
- n is the number of times interest is compounded per year (1, since it is compounded annually)
- t is the number of years

- In our case, the sum doubles in 5 years, so the final amount is 2 times the principal amount, and the number of times interest is compounded per year is 1.
- Plugging these values into the formula, we get: 2 = P(1 + r/1)^(1*5)
- Simplifying, we have: 2 = P(1 + r)^5

Step 3: Find the growth rate
- To find the growth rate (r), we need to solve the equation 2 = P(1 + r)^5 for r.
- Dividing both sides of the equation by P, we get: 2/P = (1 + r)^5
- Taking the fifth root of both sides, we have: (2/P)^(1/5) = 1 + r
- Subtracting 1 from both sides, we get: (2/P)^(1/5) - 1 = r

Step 4: Calculate the time to become 8 times
- We want to find the time it takes for the sum to become 8 times the principal amount.
- Using the formula A = P(1 + r/n)^(nt), we can set up the equation 8 = P(1 + r)^(5t), where P is the principal amount, r is the growth rate, and t is the number of years.
- Since we already know the growth rate (r), we can substitute it into the equation: 8 = P((2/P)^(1/5) - 1)^(5t)
- Simplifying, we have: 8 = (2/P)^(t/5) - 1
- Adding 1 to both sides, we get: 9 = (2/P)^(t/5)
- Taking the logarithm of both sides, we have: log(9) = (t/5) * log(2/P)
- Solving for t, we get: t = 5 * log(9) / log(2/P)
- Substituting the given values, we can calculate the time it takes for the sum to become 8 times.

Step 5: Calculate the time in hours
- The time is given in years, but we need to convert it to hours.
- Assuming there are 365 days in a year, we can