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
andrezito [222]
2 years ago
14

Solve these recurrence relations together with the initial conditions given. a) an= an−1+6an−2 for n ≥ 2, a0= 3, a1= 6 b) an= 7a

n−1−10an−2for n ≥ 2, a0= 2, a1= 1 c) an= 6an−1−8an−2for n ≥ 2, a0= 4, a1= 10 d) an= 2an−1−an−2for n ≥ 2, a0= 4, a1= 1 e) an= an−2for n ≥ 2, a0= 5, a1= -1 f) an=− 6an−1−9an−2for n ≥ 2, a0= 3, a1= -3 g) an+2 = -4an+15anfor n ≥ 0, a0= 2, a1= 8
Mathematics
1 answer:
8_murik_8 [283]2 years ago
6 0

Answer:

  • a) 3/5·((-2)^n + 4·3^n)
  • b) 3·2^n - 5^n
  • c) 3·2^n + 4^n
  • d) 4 - 3 n
  • e) 2 + 3·(-1)^n
  • f) (-3)^n·(3 - 2n)
  • g) ((-2 - √19)^n·(-6 + √19) + (-2 + √19)^n·(6 + √19))/√19

Step-by-step explanation:

These homogeneous recurrence relations of degree 2 have one of two solutions. Problems a, b, c, e, g have one solution; problems d and f have a slightly different solution. The solution method is similar, up to a point.

If there is a solution of the form a[n]=r^n, then it will satisfy ...

  r^n=c_1\cdot r^{n-1}+c_2\cdot r^{n-2}

Rearranging and dividing by r^{n-2}, we get the quadratic ...

  r^2-c_1r-c_2=0

The quadratic formula tells us values of r that satisfy this are ...

  r=\dfrac{c_1\pm\sqrt{c_1^2+4c_2}}{2}

We can call these values of r by the names r₁ and r₂.

Then, for some coefficients p and q, the solution to the recurrence relation is ...

  a[n]=pr_1^n+qr_2^n

We can find p and q by solving the initial condition equations:

\left[\begin{array}{cc}1&1\\r_1&r_2\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

These have the solution ...

p=\dfrac{a[0]r_2-a[1]}{r_2-r_1}\\\\q=\dfrac{a[1]-a[0]r_1}{r_2-r_1}

_____

Using these formulas on the first recurrence relation, we get ...

a)

c_1=1,\ c_2=6,\ a[0]=3,\ a[1]=6\\\\r_1=\dfrac{1+\sqrt{1^2+4\cdot 6}}{2}=3,\ r_2=\dfrac{1-\sqrt{1^2+4\cdot 6}}{2}=-2\\\\p=\dfrac{3(-2)-6}{-5}=\dfrac{12}{5},\ q=\dfrac{6-3(3)}{-5}=\dfrac{3}{5}\\\\a[n]=\dfrac{3}{5}(-2)^n+\dfrac{12}{5}3^n

__

The rest of (b), (c), (e), (g) are solved in exactly the same way. A spreadsheet or graphing calculator can ease the process of finding the roots and coefficients for the given recurrence constants. (It's a matter of plugging in the numbers and doing the arithmetic.)

_____

For problems (d) and (f), the quadratic has one root with multiplicity 2. So, the formulas for p and q don't work and we must do something different. The generic solution in this case is ...

  a[n]=(p+qn)r^n

The initial condition equations are now ...

\left[\begin{array}{cc}1&0\\r&r\end{array}\right] \left[\begin{array}{c}p\\q\end{array}\right] =\left[\begin{array}{c}a[0]\\a[1]\end{array}\right]

and the solutions for p and q are ...

p=a[0]\\\\q=\dfrac{a[1]-a[0]r}{r}

__

Using these formulas on problem (d), we get ...

d)

c_1=2,\ c_2=-1,\ a[0]=4,\ a[1]=1\\\\r=\dfrac{2+\sqrt{2^2+4(-1)}}{2}=1\\\\p=4,\ q=\dfrac{1-4(1)}{1}=-3\\\\a[n]=4-3n

__

And for problem (f), we get ...

f)

c_1=-6,\ c_2=-9,\ a[0]=3,\ a[1]=-3\\\\r=\dfrac{-6+\sqrt{6^2+4(-9)}}{2}=-3\\\\p=3,\ q=\dfrac{-3-3(-3)}{-3}=-2\\\\a[n]=(3-2n)(-3)^n

_____

<em>Comment on problem g</em>

Yes, the bases of the exponential terms are conjugate irrational numbers. When the terms are evaluated, they do resolve to rational numbers.

You might be interested in
a man steals 500naira from a shop,then bought an items worth 200naira collect 300naira change.how much does the shop keeper lose
Liono4ka [1.6K]
Aaaaaaaaaaaaaasssssssssssssssuiiiiiiiiiiiiiyyyyyyyyyyyyyyaaaaaaaaaaaaxrgcghgg%%%%%%%
5 0
3 years ago
Helllllllllllllllllllllllllllllllllllp
Anna11 [10]
X/4-9=5
x/4=14
x=14/4
x=7/2
x=3.5
3 0
2 years ago
What is the axis of symmetry for the function (x)=5x^2+8?
Hunter-Best [27]

Answer:

x = 0

Step-by-step explanation:

There's no horizontal shift.

8 0
3 years ago
What was your LEAST FAVORITE parts to PE this year?
VLD [36.1K]
The ones where I had to dance. For me, I am not a big fan of dancing and we had to do the la cha cha as our warm ups everyday. It was so weird. I think I rather do 5 push-ups, 10 sit-ups and jumping jacks for a warm up instead. :V
3 0
3 years ago
Read 2 more answers
A researcher wishes to be 95% confident that her estimate of the true proportion of individuals who travel overseas is within 3%
andre [41]

Answer:  a) 683   b) 1067

Step-by-step explanation:

The confidence interval for population proportion is given by :-

p\pm z_{\alpha/2}\sqrt{\dfrac{p(1-p)}{n}}

a) Given : Significance level :\alpha=1-0.95=0.05

Critical value : z_{\alpha/2}}=\pm1.96

Margin of error : E=0.03

Formula to calculate the sample size needed for interval estimate of population proportion :-

n=p(1-p)(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.2(0.8)(\dfrac{1.96}{0.03})^2=682.951111111\approx683

Hence, the required sample size would be 683 .

b) If no estimate of the sample proportion is available then the formula to calculate sample size will be :-

n=0.25(\dfrac{z_{\alpha/2}}{E})^2\\\\=0.25(\dfrac{1.96}{0.03})^2=1067.11111111\approx1067

Hence, the required sample size would be 1067 .

3 0
2 years ago
Other questions:
  • Floral designers often create arrangements where the flower height to container ratio is 5 to 3. A certain museum wishes to crea
    15·1 answer
  • Aden is going to purchase sod for his backyard. How many square feet of sod will he need? There is a picture of a square with 15
    8·1 answer
  • How do you represent the five in 6.75
    14·1 answer
  • How to find slope and y intercept 6x-5y=15?
    6·1 answer
  • PLEASE HELP!!!!
    10·1 answer
  • micchel and jennefer each worked 25 hours last week. micchel eared $10 per hour jennifer earned $12 per hour how much more did j
    5·1 answer
  • Say you flip a coin seven times. What is the probability the number of heads will be even?
    12·1 answer
  • Every Saturday, Heather goes mountain biking. It takes Heather 32 minutes to bike 6 miles. The distance is a function of the tim
    14·2 answers
  • Which mathematical property is demonstrated below? 3 + 0 = 3
    11·2 answers
  • What is the quadratic regression equation that fits these data
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!