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The problem involves compound interest and asks us to compare the amount of money invested over various years. Let's break it down:

### Key Information:
1. Let \( P \) be the principal (initial sum invested).
2. Let \( r \) be the compound interest rate per year.
3. At the end of year \( n \), the amount will be \( P(1 + r)^n \).

### Given Condition:
- The amount at the end of the 7th year is twice the amount at the end of the 2nd year.
  
This gives us the equation:
\[
P(1 + r)^7 = 2 \cdot P(1 + r)^2
\]

We can simplify this equation by canceling \( P \) from both sides (since \( P \neq 0 \)):

\[
(1 + r)^7 = 2 \cdot (1 + r)^2
\]

Now, divide both sides by \( (1 + r)^2 \):

\[
(1 + r)^5 = 2
\]

Thus, \( (1 + r)^5 = 2 \).

### Now, the problem asks for the amount at the end of the 64th year compared to the amount at the end of the 49th year.

The amount at the end of the 64th year is:
\[
A_{64} = P(1 + r)^{64}
\]
The amount at the end of the 49th year is:
\[
A_{49} = P(1 + r)^{49}
\]

We need to find how many times \( A_{64} \) is of \( A_{49} \). This is given by:
\[
\frac{A_{64}}{A_{49}} = \frac{P(1 + r)^{64}}{P(1 + r)^{49}} = (1 + r)^{64 - 49} = (1 + r)^{15}
\]

But from the earlier equation \( (1 + r)^5 = 2 \), we can express \( (1 + r)^{15} \) as:
\[
(1 + r)^{15} = \left( (1 + r)^5 \right)^3 = 2^3 = 8
\]

### Final Answer:
The amount at the end of the 64th year is **8 times** the amount at the end of the 49th year.

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Would you like more details or have any questions?

Here are 5 related questions:
1. How does the compound interest formula change if interest is compounded more frequently, like monthly?
2. What happens to the comparison of amounts if the interest rate \( r \) is increased?
3. Can the equation \( (1 + r)^5 = 2 \) be solved for \( r \), and what would that value represent?
4. How does simple interest differ from compound interest in this kind of problem?
5. What if the time period was halved, how would that impact the final comparison?

**Tip**: When dealing with compound interest, always try to express all terms in terms of powers of \( (1 + r) \) for easier comparison.

If a certain sum, invested under compound interest, amounts to twice as much at the end of the seventh year as it would at the end of the second year, then the amount at the end of the 64th year will be how many times of the amount that was at the end of the 49th year?

Solve a compound interest problem where the amount doubles after 7 years, and find out how many times the amount at year 64 is compared to year 49. This problem explores algebraic concepts such as exponential growth and laws of exponents. Suitable for students in Grades 10-12.

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