1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Darya [45]
3 years ago
9

Let an = –3an-1 + 10an-2 with initial conditions a1 = 29 and a2 = –47. a) Write the first 5 terms of the recurrence relation. b)

Solve this recurrence relation. Show your reasoning. c) Using the explicit formula you found in part b, evaluate a5. You must show that you are using the equation from part b.
Mathematics
1 answer:
zlopas [31]3 years ago
6 0

We can express the recurrence,

\begin{cases}a_1=29\\a_2=-47\\a_n=-3a_{n-1}+10a_{n-2}7\text{for }n\ge3\end{cases}

in matrix form as

\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}\begin{bmatrix}a_{n-1}\\a_{n-2}\end{bmatrix}

By substitution,

\begin{bmatrix}a_{n-1}\\a_{n-2}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}\begin{bmatrix}a_{n-2}\\a_{n-3}\end{bmatrix}\implies\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}^2\begin{bmatrix}a_{n-2}\\a_{n-3}\end{bmatrix}

and continuing in this way we would find that

\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}\begin{bmatrix}a_2\\a_1\end{bmatrix}

Diagonalizing the coefficient matrix gives us

\begin{bmatrix}-3&10\\1&0\end{bmatrix}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}-5&0\\0&2\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}

which makes taking the (n-2)-th power trivial:

\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}-5&0\\0&2\end{bmatrix}^{n-2}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}

\begin{bmatrix}-3&10\\1&0\end{bmatrix}^{n-2}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}(-5)^{n-2}&0\\0&2^{n-2}\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}

So we have

\begin{bmatrix}a_n\\a_{n-1}\end{bmatrix}=\begin{bmatrix}-5&2\\1&1\end{bmatrix}\begin{bmatrix}(-5)^{n-2}&0\\0&2^{n-2}\end{bmatrix}\begin{bmatrix}-5&2\\1&1\end{bmatrix}^{-1}\begin{bmatrix}a_2\\a_1\end{bmatrix}

and in particular,

a_n=\dfrac{29\left(2(-5)^{n-1}+5\cdot2^{n-1}\right)-47\left(-(-5)^{n-1}+2^{n-1}\right)}7

a_n=\dfrac{105(-5)^{n-1}+98\cdot2^{n-1}}7

a_n=15(-5)^{n-1}+14\cdot2^{n-1}

\boxed{a_n=-3(-5)^n+7\cdot2^n}

You might be interested in
Can someone give me the answer and solution plz
Masteriza [31]

Answer: May.

Step-by-step explanation:

The chart has its highest rabbit population at January and October, so it would not be either of those because it is asking for the smallest, checking the red line and the numbers next to it, may would have the absolute smallest population around 450.

5 0
3 years ago
(The question is in the photo)
Naddik [55]

Answer:

6.99

Step-by-step explanation:

11.99+2.99+5.99=20.97

20.97/3=6.99

6 0
3 years ago
Read 2 more answers
What is the measure of angle x ?
Arada [10]

Answer:

It should either be 69 or 70

Step-by-step explanation:

8 0
2 years ago
Help me plssssssssssss....<br>I will give brainlist <br>​
olya-2409 [2.1K]

Answer:

C) 1/3

Step-by-step explanation:

First look at equation: 3x+y=1

re-write as y=mx+b

y=-3x+1

so the slope of this equation i s-3

so if a line is perpendicular to the above equation,

its slope will be reciprocal negative which is 1/3

6 0
2 years ago
What is the y-intercept, axis of symmetry and vertex of the following function.
pshichka [43]
The y intercept is always c (-5) and to the axis of symmetry u do -b/2a which is -1 and the vertex is (-1,10) because u pug in -1 into the equation. lmk if it’s right!
6 0
2 years ago
Other questions:
  • Plz help I am giving away 20 points!!!!!!!!
    14·2 answers
  • Nick bought apples at a farmers market where 5 apples cost $4.45.
    14·2 answers
  • 62 POINTS!!!
    7·2 answers
  • Solve the system of equations.<br> - 5x + 2y = 9
    5·1 answer
  • PLS HELP ASAP
    7·1 answer
  • A survey of a group of seventh graders and a group of teachers at a local middle school asked how many siblings they each have.
    11·2 answers
  • Whats 78 × 4 × 56 × 7 ×2 <br><br><br> This is random just for fun:)
    14·2 answers
  • -2(3x-7) <br> Show your work.
    12·2 answers
  • It cost Brody $9.80 to send 196 text messages. How many text messages did he send if he spent $4.20?
    10·1 answer
  • PLEASE HELP!! find the value of x. attached image.
    9·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!