Answer :
Final answer:
The problem can be solved by adjusting the interest rate for semiannual compounding, putting values in the formula for compound interest, and then utilizing the properties of logarithms to solve for the required time period. The final answer will be the number of six month periods, so it should be divided by 2 to get the number of years.
Explanation:
The question is about the calculation of how many years will it take for a certain amount of money to increase to a given sum, using a particular interest rate, with the interest compounded semiannually. This is typically solved by using the formula for compound interest and the properties of logarithms.
First, we need to adjust our interest rate and our compounding period. Since the interest rate of 6% is compounded semiannually, it becomes 3% or 0.03 per compound period.
We are given the formula F = P(1+i)^n, where F is the future value, P is the principal amount, i is the interest rate per period, and n is the number of periods. Here, F = 20000, P = 1000, and i = 0.03. We are tasked with finding n.
So, we start with 20000 = 1000(1 + 0.03)^n. We can simplify this to 20 = (1.03)^n. We then use logarithmic function properties to solve for n. Using logarithms allows us to bring down the exponent as a multiplier. By applying the logarithm function to both sides, we get log(20) = n * log(1.03).
We solve for n by dividing both sides of this equation by log(1.03): n = log(20) / log(1.03)
Calculating these logarithms gives us the value of n, which will be the number of six month periods it will take for 1000 taka to grow to 20000. To find out the number of years, we should divide n by 2.
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