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3 years ago

5

Regina took out a 30-year loan for $110,000 at an APR of 9.6%, compounded

2 answers:

sergeinik [125]3 years ago

8 0

Answer: $82,975.68 APEX

GREYUIT [131]3 years ago

7 0

Answer: 82975.68 apex

Step-by-step explanation:

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13.10. Suppose that a sequence (ao, a1, a2, ) of real numbers satisfies the recurrence relation an -5an-1+6an-20 for all n> 2

Gwar [14]

a. This recurrence is of order 2.

b. We're looking for a function $A(x)$ such that

$A(x)=\displaystyle\sum_{n=0}^\infty a_nx^n$

Take the recurrence,

$\begin{cases}a_0=a_0\\a_1=a_1\\a_n-5a_{n-1}+6a_{n-2}=0&\text{for }n\ge2\end{cases}$

Multiply both sides by $x^{n-2}$ and sum over all integers $n\ge2$ :

$\displaystyle\sum_{n=2}^\infty a_nx^{n-2}-5\sum_{n=2}^\infty a_{n-1}x^{n-2}+6\sum_{n=2}^\infty a_{n-2}x^{n-2}=0$

Pull out powers of $x$ so that each summand takes the form $a_kx^k$ :

$\displaystyle\frac1{x^2}\sum_{n=2}^\infty a_nx^n-\frac5x\sum_{n=2}^\infty a_{n-1}x^{n-1}+6\sum_{n=2}^\infty a_{n-2}x^{n-2}=0$

Now shift the indices and add/subtract terms as needed to get everything in terms of $A(x)$ :

$\displaystyle\frac1{x^2}\left(\sum_{n=0}^\infty a_nx^n-a_0-a_1x\right)-\frac5x\left(\sum_{n=0}^\infty a_nx^n-a_0\right)+6\sum_{n=0}^\infty a_nx^n=0$

$\displaystyle\frac{A(x)-a_0-a_1x}{x^2}-\frac{5(A(x)-a_0)}x+6A(x)=0$

Solve for $A(x)$ :

$A(x)=\dfrac{a_0+(a_1-5a_0)x}{1-5x+6x^2}\implies\boxed{A(x)=\dfrac{a_0+(a_1-5a_0)x}{(1-3x)(1-2x)}}$

c. Splitting $A(x)$ into partial fractions gives

$A(x)=\dfrac{2a_0-a_1}{1-3x}+\dfrac{3a_0-a_1}{1-2x}$

Recall that for $|x|$ , we have

$\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n$

so that for $|3x|$ and $|2x|$ , or simply $|x|$ , we have

$A(x)=\displaystyle\sum_{n=0}^\infty\bigg((2a_0-a_1)3^n+(3a_0-a_1)2^n\bigg)x^n$

which means the solution to the recurrence is

$\boxed{a_n=(2a_0-a_1)3^n+(3a_0-a_1)2^n}$

d. I guess you mean $a_0=2$ and $a_1=5$ , in which case

$\boxed{\begin{cases}a_0=2\\a_1=5\\a_2=13\\a_3=35\\a_4=97\\a_5=275\end{cases}}$

e. We already know the general solution in terms of $a_0$ and $a_1$ , so just plug them in:

$\boxed{a_n=2^n+3^n}$

8 0

3 years ago

What is the probability that in a group of 2 random digits both will be odd

Slav-nsk [51]

There will be an 1/2 probability

8 0

3 years ago

Are triangles DEF and LNM similar if DF = 8 and LM = 4?triangles LNM and DEF with angles N and E marked with the right angle sym

Sophie [7]

Answer:

Option D) No, there is not enough information

Step-by-step explanation:

We are given the following in the question:

A triangles DEF and triangle LNM

$\angle N = \angle E = 90^\circ\\DF = 8\\LM = 4$

Similarity of triangle can be proved in the following manner:

AA
SSS
SAS

Since from the given information, it is not possible to prove the similarity of the triangles due to incomplete information, the triangles cannot be proved similar by any postulate.

Thus, the correct answer is

Option D) No, there is not enough information

5 0

3 years ago

Read 2 more answers

Which of the following coordinates exits on the line y=3x+2

raketka [301]

It is B

-2 =3(-1) + 1

3 0

3 years ago

Function f(x) = 4x - 6 with a domain of (-3,5), identify the range

VLD [36.1K]

Answer:

(-18,5)

Step-by-step explanation:

4 0

4 years ago