Business: Pharoah Inc. loans money to John Kruk Corporation in the amount of $976,000. Pharoah accepts an 8% note due in 7 years with interest payable semiannually. After 2 years (and receipt of interest for 2 years), Pharoah needs money and therefore sells the note to Chicago National Bank, which demands interest on the note of 10% compounded semiannually. What is the amount Pharoah will receive on the sale of the note

Pharoah Inc. loans money to John Kruk Corporation

swaggernas

18.04.2021

$‭900,635

Explanation:

Amount Pharaoh will receive is:

= Present value of the interest payments + Present value of the note

2 years have gone by which leaves 5 years.

Period = 5 * 2 = 10 semi annual periods

Periodic interest = 8% / 2 = 4%

Periodic discount = 10% / 2 = 5% per period

Interest payment = 976,000 * 4%

= $‭39,040‬

Amount to be received:

= (‭39,040‬ * Present value interest factor of annuity, 5%, 10 periods) + 976,000/(1 + 5%)¹⁰

= ‭(‭39,040‬ * 7.7217) + 599,179.34

= $‭900,635

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Kstearns694

28.12.2022

Not sure why my previous answer was deleted...

If you substitute $y=\frac1{\sin^2x}=\csc^2x$ , then $x\to0$ gives $y\to+\infty$ , so the limit is equal in value to

$\displaystyle\lim_{y\to\infty}\left(1^y+2^y+\cdots+n^y\right)^{1/y}$

Since $e^{\ln x}=x$ , we can write

$\left(1^y+2^y+\cdots+n^y\right)^{1/y}=e^{\ln(1^y+2^y+\cdots+n^y)^{1/y}=e^{\ln(1^y+2^y+\cdots+n^y)/y}$

Because e^x is continuous at all , we can pass the limit to the argument of the exponential function. That is,

$\displaystyle\lim_{x\to\infty}e^{f(x)}=e^{\lim\limits_{x\to\infty}f(x)}$

so that the limit we're interested in is equal to

$e^{\lim\limits_{y\to\infty}\ln(1^y+2^y+\cdots+n^y)/y}$

For all natural numbers , we have

$1^y+2^y+\cdots+n^y\le n^y+n^y+\cdots+n^y=n\cdot n^y=n^{y+1}$

$\displaystyle\lim_{y\to\infty}\frac{\ln(1^y+2^y+\cdots+n^y)}y=\lim_{y\to\infty}\frac{\ln n^{y+1}}y=\ln n\cdot\lim_{y\to\infty}\frac{y+1}y=\ln n$

and this makes the overall limit take on a value of

$e^{\ln n}=\boxed n$

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