Let's solve the problem step-by-step.
First, identify the given information:
1. Initial deposit: \[tex]$5
2. Balance doubles every week.
3. Goal: \$[/tex]1,280
Based on the given information, the balance of the fund grows following an exponential pattern where each term is double the previous term. The initial value is \[tex]$5.
The general form of the equation for exponential growth is:
\[ \text{Initial amount} \times (\text{base})^x = \text{final amount} \]
where:
- The initial amount is \$[/tex]5.
- The growth factor (base) is 2 because the balance doubles each week.
- The final amount is \[tex]$1,280.
- \( x \) is the number of weeks.
Substituting the given values into the equation, we get:
\[ 5 \times (2)^x = 1,280 \]
Now we need to solve for \( x \) to find the number of weeks it takes for the balance to reach \$[/tex]1,280.
To solve the equation:
[tex]\[ 5 \times (2)^x = 1,280 \][/tex]
First, divide both sides of the equation by 5:
[tex]\[ (2)^x = \frac{1,280}{5} \][/tex]
[tex]\[ (2)^x = 256 \][/tex]
Next, recognize that 256 is a power of 2, specifically:
[tex]\[ 256 = 2^8 \][/tex]
Therefore:
[tex]\[ (2)^x = (2)^8 \][/tex]
By comparing the exponents of the same base, we get:
[tex]\[ x = 8 \][/tex]
So, the correct equation and value of [tex]\( x \)[/tex] is:
[tex]\[ 5(2)^x = 1,280 \; \text{and} \; x = 8 \][/tex]
Based on this solution, the correct option is:
A. [tex]\( 5(2)^x = 1,280 \; \text{;}\; x = 8 \)[/tex]
Thus, it will take 8 weeks for the balance of the fund to reach \$1,280.
