Problem 8:

The sequence provided is: $1, \sqrt{2}, 2, 2\sqrt{2}, \dots$

The formula given is: $a_n = (\sqrt{2})^{n-1}$.

Checking the Formula:

• For $n = 1$: $a_1 = (\sqrt{2})^{1-1} = (\sqrt{2})^0 = 1$ ✅
• For $n = 2$: $a_2 = (\sqrt{2})^{2-1} = (\sqrt{2})^1 = \sqrt{2}$ ✅
• For $n = 3$: $a_3 = (\sqrt{2})^{3-1} = (\sqrt{2})^2 = 2$ ✅
• For $n = 4$: $a_4 = (\sqrt{2})^{4-1} = (\sqrt{2})^3 = 2\sqrt{2}$ ✅

Since the formula $a_n = (\sqrt{2})^{n-1}$ accurately represents the sequence, the answer is True.

Problem 9:

Brady's inheritance problem involves compound interest, calculated using the formula:

$A = P \left( 1 + \frac{r}{n} \right)^{nt}$

Where:

• $P = 20,000$ (initial principal),
• $r = 0.046$ (annual interest rate),
• $n = 12$ (compounding periods per year),
• $t = 15$ (time in years, from age 5 to 20).

Calculation:

$A = 20000 \left( 1 + \frac{0.046}{12} \right)^{12 \times 15}$

$A = 20000 \left( 1 + 0.0038333 \right)^{180}$

$A = 20000 \left( 1.0038333 \right)^{180}$

$A \approx 20000 \times 2.1138357$

$A \approx 42,276.71$

Brady's inheritance will be worth approximately $42,276.71 when he turns 20.

Do you have further questions or need additional clarifications?

Related Questions:

1. Can you explain why compound interest grows faster than simple interest?
2. How does the number of compounding periods affect the final amount?
3. What would happen if the interest rate were compounded yearly instead of monthly?
4. How is the sequence formula $(\sqrt{2})^{n-1}$ derived?
5. What happens to Brady's inheritance if the rate changes to 5%?

Tip:

When solving compound interest problems, always confirm the compounding frequency, as it can significantly affect the growth of investments.