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A zero-coupon bond sells for $ 250. At maturity in 16 years, it will be redeemed for $ 1,000. What approximate annual rate of growth does this represent?
Using the future value formula: \(FV = PV \times (1+r)^n\). So \(1000 = 250\times(1+r)^{16}\). \((1+r)^{16} = 4\). \(1+r = 4^{1/16} = 4^{0.0625}\). \(\ln(1+r) = \frac{\ln 4}{16} = \frac{1.386}{16} = 0.0866\). \(1+r = e^{0.0866} \approx 1.0905\). So \(r \approx 9\%\). Wait: \(4^{1/16}\): \(\log(4)/16 = 0.602/16=0.0376\), antilog \(=1.089\), so \(r\approx 9\%\) = option (B). Re-examining: \(4^{0.0625}=1.0905\Rightarrow r\approx9\%\). Answer: option (B) = 9 percent.
Let integer $v > 1$ be a perfect square, so $v = k^2$ for some integer $k \geq 2$, and $\sqrt{v} = k$. Which of the following expressions must also be a perfect square? Select all that apply.
Since $v = k^2$ and $\sqrt{v} = k$:
Option A โ $81v$: $= 81k^2 = (9k)^2$. Always a perfect square. โ
Option B โ $25v + 10\sqrt{v} + 1$: $= 25k^2 + 10k + 1 = (5k+1)^2$. Always a perfect square. โ
Option C โ $4v^2 + 4\sqrt{v} + 1$: $= 4k^4 + 4k + 1$. Test $k = 2$: $4(16) + 4(2) + 1 = 73$. $\sqrt{73}$ is not an integer. Not always a perfect square. โ
Option D โ $v + 2\sqrt{v} + 1$: $= k^2 + 2k + 1 = (k+1)^2$. Always a perfect square. โ
Correct answers: A, B, and D (indices 0, 1, 3).
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