Answer:
a) ![A(t) = 17500(1.0053)^{12t}](https://tex.z-dn.net/?f=A%28t%29%20%3D%2017500%281.0053%29%5E%7B12t%7D)
b) The balance after 8 years will be of $29,069.
Step-by-step explanation:
Compound interest:
The compound interest formula is given by:
![A(t) = P(1 + \frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20P%281%20%2B%20%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
Where A(t) is the amount of money after t years, P is the principal(the initial sum of money), r is the interest rate(as a decimal value), n is the number of times that interest is compounded per year and t is the time in years for which the money is invested or borrowed.
a)
Loan of $17,500 means that ![P = 17500](https://tex.z-dn.net/?f=P%20%3D%2017500)
6.4% interest rate means that ![r = 0.064](https://tex.z-dn.net/?f=r%20%3D%200.064)
Compounded monthly means that
. So
![A(t) = P(1 + \frac{r}{n})^{nt}](https://tex.z-dn.net/?f=A%28t%29%20%3D%20P%281%20%2B%20%5Cfrac%7Br%7D%7Bn%7D%29%5E%7Bnt%7D)
![A(t) = 17500(1 + \frac{0.064}{12})^{12t}](https://tex.z-dn.net/?f=A%28t%29%20%3D%2017500%281%20%2B%20%5Cfrac%7B0.064%7D%7B12%7D%29%5E%7B12t%7D)
![A(t) = 17500(1.0053)^{12t}](https://tex.z-dn.net/?f=A%28t%29%20%3D%2017500%281.0053%29%5E%7B12t%7D)
b)
This is A(8). Then
![A(8) = 17500(1.0053)^{12*8} = 29069](https://tex.z-dn.net/?f=A%288%29%20%3D%2017500%281.0053%29%5E%7B12%2A8%7D%20%3D%2029069)
The balance after 8 years will be of $29,069.
The next logical letter would be w.
This pattern starts at the first letter, then the last letter, then the second letter, then the second to last, the third, third to last, and so on.
Answer:
![\frac{13-7\sqrt{3}}{2}](https://tex.z-dn.net/?f=%5Cfrac%7B13-7%5Csqrt%7B3%7D%7D%7B2%7D)
Step-by-step explanation:
We need to rationalize the denominator of
. For rationalizing we multiply the equation by ![\frac{4-2\sqrt{3}}{4-2\sqrt{3}}](https://tex.z-dn.net/?f=%5Cfrac%7B4-2%5Csqrt%7B3%7D%7D%7B4-2%5Csqrt%7B3%7D%7D)
So, solving
![= \frac{5-\sqrt{3}}{4+2\sqrt{3}}*\frac{4-2\sqrt{3}}{4-2\sqrt{3}} \\=\frac{(5-\sqrt{3})(4-2\sqrt{3})}{4+2\sqrt{3}*4-2\sqrt{3}}\\=\frac{(5-\sqrt{3})(4-2\sqrt{3})}{(4)^2-(2\sqrt{3})^2}\\= \frac{5(4-2\sqrt{3})-\sqrt{3}(4-2\sqrt{3})}{16-(4*3)}\\=\frac{20-10\sqrt{3}-4\sqrt{3}+2*3}{16-12}\\=\frac{20+6-14\sqrt{3}}{4}\\=\frac{26-14\sqrt{3}}{4}\\= \frac{2(13-7\sqrt{3})}{4}\\=\frac{13-7\sqrt{3}}{2}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B5-%5Csqrt%7B3%7D%7D%7B4%2B2%5Csqrt%7B3%7D%7D%2A%5Cfrac%7B4-2%5Csqrt%7B3%7D%7D%7B4-2%5Csqrt%7B3%7D%7D%20%5C%5C%3D%5Cfrac%7B%285-%5Csqrt%7B3%7D%29%284-2%5Csqrt%7B3%7D%29%7D%7B4%2B2%5Csqrt%7B3%7D%2A4-2%5Csqrt%7B3%7D%7D%5C%5C%3D%5Cfrac%7B%285-%5Csqrt%7B3%7D%29%284-2%5Csqrt%7B3%7D%29%7D%7B%284%29%5E2-%282%5Csqrt%7B3%7D%29%5E2%7D%5C%5C%3D%20%5Cfrac%7B5%284-2%5Csqrt%7B3%7D%29-%5Csqrt%7B3%7D%284-2%5Csqrt%7B3%7D%29%7D%7B16-%284%2A3%29%7D%5C%5C%3D%5Cfrac%7B20-10%5Csqrt%7B3%7D-4%5Csqrt%7B3%7D%2B2%2A3%7D%7B16-12%7D%5C%5C%3D%5Cfrac%7B20%2B6-14%5Csqrt%7B3%7D%7D%7B4%7D%5C%5C%3D%5Cfrac%7B26-14%5Csqrt%7B3%7D%7D%7B4%7D%5C%5C%3D%20%5Cfrac%7B2%2813-7%5Csqrt%7B3%7D%29%7D%7B4%7D%5C%5C%3D%5Cfrac%7B13-7%5Csqrt%7B3%7D%7D%7B2%7D)