An investment of $1 was put in an account. Every 8 years, the money doubled. No additional money was added to the account. Would the expression $1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ correctly represent how much was in the investment account after 48 years? Explain.

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nshupe54 nshupe54

06-11-2026
Mathematics

Answered

An investment of $1 was put in an account. Every 8 years, the money doubled. No additional money was added to the account. Would the expression $1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ correctly represent how much was in the investment account after 48 years? Explain.

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T19:18:33-05:00

The investment doubles every $8$ years, and in $48$ years, there are $48/8 = 6$ doubling periods.
The amount after $48$ years is represented by $1 \times 2^6$.
The expression $1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ is equivalent to $1 \times 2^6$.
The expression correctly represents the investment after $48$ years. $\boxed{\text{Yes}}$

Explanation

Analyze the Investment Growth

An investment of $\$1$ doubles every $8$ years. To determine the amount in the account after $48$ years, we calculate how many times the money doubles. The number of doubling periods $n$ is given by the total time divided by the time it takes for one doubling:
\[n = \frac{48 \text{ years}}{8 \text{ years/period}} = 6 \text{ periods}\]
The growth of the investment follows an exponential model $A = P \cdot 2^n$, where $P$ is the initial principal and $n$ is the number of doubling periods. Substituting our known values $P = \$1$ and $n = 6$, the amount after $48$ years is:
\[A = 1 \cdot 2^6 = 2^6\]

Evaluate the Expression

The provided expression is $1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$. We can rewrite this as:
\[1 \times 2^6\]
Since we established that the amount after $48$ years is $1 \cdot 2^6$, we see that the expression correctly matches the growth model. The number of times the investment doubles is exactly $6$, which corresponds to the six factors of $2$ in the expression provided. Thus, the expression correctly represents the total amount in the account after $48$ years.

Conclusion

The expression $1 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ accurately calculates the amount after $48$ years because it represents the initial $\$1$ being doubled $6$ times over a span of $48$ years. The final amount in the account is $\$64$. Therefore, the expression is correct. $\boxed{\text{Yes}}$

Examples

Understanding compound growth is essential in finance, such as when calculating interest in a savings account or the growth of an investment over time. For example, if you place $\$100$ in an account that doubles every $10$ years, you can use the same logic to calculate the value after $30$ years: $100 \times 2^3 = 100 \times 8 = \$800$. This helps in planning for long-term financial goals and understanding how exponential growth affects your savings.