Question : A certain sum becomes $k$ times in 5 years at compound interest. In 30 years, it will become how many times?
Option 1: $5k$
Option 2: $6k$
Option 3: $k^6$
Option 4: $k^5$
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Correct Answer: $k^6$
Solution : Let the original sum as $P$, the annual interest rate as $R$, and the amount after $n$ years is $kP$. So, $kP=P(1 + \frac{R}{100})^5$ ⇒ $k=(1 + \frac{R}{100})^5$ -----------------(i) After 30 years, $A=P(1 + \frac{R}{100})^{30}$ ⇒ $A=P({(1 + \frac{R}{100})^5})^6$ From equation (i), we get, $A = Pk^6$ ⇒ $\frac{A}{P}=k^6$ Hence, the correct answer is $k^6$.
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