(a) There are 2 sequences of length 1 that fit the bill: 0 and 1. So if [tex]b_n[/tex] is the number of permissible sequences, then [tex]b_1=2[/tex].
To the sequence 0, we can attach either 0 or 1 at the end, while the sequence 1 must be followed by 0. Then the permissible sequences of length 2 are 00, 01, and 10, so [tex]b_2=3[/tex].
Now consider a sequence of length n + 1, of which there are [tex]b_{n+1}[/tex].
• If the last digit is 0, then we got this sequence by simply joining 0 to one of [tex]b_n[/tex] sequences. In other words, there are [tex]b_n[/tex] permissible sequences of length n + 1 that end in 0.
• If the last digit is 1, then the previous digit must have been 0. Put another way, we join 01 to a permissible sequence of length n - 1 (there are [tex]b_{n-1}[/tex] of them). So there are [tex]b_{n-1}[/tex] permissible sequences of length n + 1 that end in 1.
These cases are mutually exclusive, so the number of (n + 1)-length permissible sequences is given by
[tex]\begin{cases}b_1 = 2 \\ b_2 = 3 \\ b_{n+1} = b_n + b_{n-2} & \text{for }n\ge2\end{cases}[/tex]
(b) The reasoning for a permissible ternary sequence is similar. Let [tex]t_n[/tex] be the number of sequences of length n not containing 11.
We first note that [tex]t_1 = 3[/tex], since we can have 0, 1, or 2; and [tex]t_2 = 8[/tex], since we can have 00, 01, 02, 10, 12, 20, 21, or 22.
Consider a sequence of length n + 1 ([tex]t_{n+1}[/tex] of these).
• If the last digit is 0, then we joined 0 to a permissible sequence of length n. So there are [tex]t_n[/tex] permissible sequences of length n + 1 ending in 0.
• If the last digit is 1, then the last two digits must be either 01 or 21. Since there are 2 choices for the n-th digit, there are [tex]2t_{n-1}[/tex] permissible sequences of length n + 1 ending in 1.
• If the last digit is 2, then we essentially have the same situation as if the sequence ended in 0, so there are [tex]t_n[/tex] sequences of length n + 1 ending in 2.
Then the recurrence for ternary permissible sequences is
[tex]\begin{cases}t_1 = 3 \\ t_2 = 8 \\ t_{n+1} = 2t_n + 2t_{n-1} & \text{for }n\ge2\end{cases}[/tex]
pls help i will give brainlinest (please explain how to do)
Hello! The missing number is -1.
Notice that we add 1 every time:
-3+1=-2
-2+1=-1
-1+1=0
Therefore, the missing number is -1.Hope this helps!
~Just a felicitous girlie
#HaveAnAwesomeDay
[tex]SilentNature :)[/tex]
The graph shows the number of minutes you and your brother spend training your dogs for an upcoming competition. You spend___more minutes trading your dog each day than your brother.
Answer:
It is 15 but I don't know what the steps are since I took a test on this question and failed to get the right answer.
Here are three different recipes for cookies.
Recipe 1: Mix 2 cups of chocolate chips with 6 cups of cookie batter.
Recipe 2: Mix 3 cups of chocolate chips with 7 cups of cookie batter.
Recipe 3: Mix 4 cups of chocolate chips with 12 cups of cookie batter.
Which two cookie recipes will make the same cookies?
Answer:
Recipes 1 and 3 will make the same amount
4 plums and 7 bananas
Can someone please help with this?
Answer:
I was learning that to but it is way to blurred out and I can not see the text.
You need to make a clear picture if you need answers or no one can help you because we don't know what it say's.
Step-by-step explanation:
(-5/6) + (-2 5/8) equals?
Jane watched her brother Peter build something that looked like stairs out of blocks. He put a block, then a tower of 2 blocks next to it, then a tower of 3 blocks… If he continues the pattern, how tall will the tallest tower be if there are 200 blocks in the kit?
Answer:
1,200
Step-by-step explanation:
3x2=6 and 6x200=1,200
please help, this is an algebra 2 question :) I attached the problem in a photo - simplify the following expression
Answer:
[tex]\frac{x^2+19x-56}{4x^2-32x}[/tex]
Step-by-step explanation:
To add two fractions we need a common denominator. Just by looking at the two, there are no common factors. This means the denominator would be the product of the two: 4x(x-8). In order to obtain this, we multiply the first fraction by (x-8)/(x-8) - 1, and the second by 4x/4x - 1:
[tex]\frac{x+7}{4x} *\frac{x-8}{x-8} +\frac{5}{x-8}*\frac{4x}{4x}[/tex]
We can use the distributive property to simplify:
[tex]\frac{x^2-x-56}{4x(x-8)} +\frac{20x}{4x(x-8)}[/tex]
The sum of the two is:
[tex]\frac{x^2+19x-56}{4x^2-32x}[/tex]
Tìm giá trị của Lim (đề bài trong ảnh)
First, rewrite
[tex]\dfrac{x^2+5x+4}{x^2-3x+7} = \dfrac{(x^2-3x+7)+(8x-3)}{x^2-3x+7} = 1 + \dfrac{8x-3}{x^2-3x+7}[/tex]
Next, we have
[tex]\dfrac{x^2-3x+7}{8x-3} = \dfrac x8 - \dfrac{21}{64} + \dfrac{385}{512x-192}[/tex]
so we can further rewrite
[tex]\dfrac{x^2+5x+4}{x^2-3x+7} = 1 + \dfrac1{\frac x8 - \frac{21}{64} + \frac{385}{512x-192}}[/tex]
The idea behind all this is to get the limand into a form resembling the limit definition of the constant e, which is
[tex]\displaystyle e = \lim_{x\to\infty} \left(1+\frac1x\right)^x[/tex]
The next step is to make the substitution,
[tex]y = \dfrac x8 - \dfrac{21}{64} + \dfrac{385}{512x-192}[/tex]
Solving for x gives two solutions,
[tex]x = \dfrac{8y+3\pm\sqrt{64y^2+36y-19}}2[/tex]
but we want to have y approaching infinity, just like in the definition for e, which is achieved only by the solution with the positive square root. (You can check for yourself that the other solution instead approaches 3/4.)
So, our limit is equivalent to
[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac{8y+3+\sqrt{64y^2+36y-19}}2}[/tex]
[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac{8y+3+8y\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}}2}[/tex]
[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{4y\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)+\frac32}[/tex]
[tex]\displaystyle \lim_{y\to\infty} \left(\left(1+\frac1y\right)^y\right)^{4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)+\frac32}[/tex]
[tex]\displaystyle \lim_{y\to\infty} \left(\left(1+\frac1y\right)^y\right)^{4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right)} \cdot \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac32}[/tex]
Next, we use the fact that
[tex]\displaystyle \lim_{x\to\infty} f(x)^{g(x)} = \left( \lim_{x\to\infty}f(x)\right)^{\lim\limits_{x\to\infty}g(x)}[/tex]
Now, the "base" limit is e, while the "exponent" limit is
[tex]\displaystyle \lim_{y\to\infty} 4\left(1+\sqrt{1+\frac{36}{64y}-\frac{19}{64y^2}}\right) = 4(1+\sqrt{1+0+0}) = 8[/tex]
and the remaining limit is simply
[tex]\displaystyle \lim_{y\to\infty} \left(1+\frac1y\right)^{\frac32} = (1+0)^{\frac32} = 1[/tex]
So, we end up with
[tex]\displaystyle \lim_{x\to\infty} \left(\frac{x^2+5x+4}{x^2-3x+7}\right)^x = e^8 \cdot 1 = \boxed{e^8}[/tex]
If I have 4 types of wood and 5 types of other materials, what is most likely.
Answer:
Well I'm not sure what your answer choices are but based on the question I can tell you this:
If you have 4 wood and 5 other materials then you have 9 total materials.
So if you have 4 wood then you have a 4/9 chance of choosing wood.
And you have a 5/9 chance of choosing another material
So based on that, you would be more likely to choose another material, that is not wood.
I’m not quite understanding this, it’s just not setting in my brain (I’m just now starting in this math class)
Anyone who can help me, thank you I appreciate you, a lot!
I need part a and b
The question has to be worked out and if anyone doesn’t care to explain it to me! :)
Answer:
Step-by-step explanation:
Let x be the number of hours worked
Let C = the charge made by the first company
Let c = the charge made by the second company
You want to know when the costs will be =.
Equations
C = 376 + 12x
c = 280 + 15x
Solution
Make the two right sides equal
376 + 12x = 280 + 15x Subtract 12x from both sides
376 = 280 + 15x - 12x Combine like terms on the right
376 = 280 + 3x Subtract 280 from both sides
376 - 280 = 3x Combine
96 = 3x Divide both sides by 3
32 = x
Each company would have to work 32 hours to get the same amount to be charged.
C = 376 + 12x
C = 376 + 12*32
C = 376 + 384
C = 760
The amount each would charge is 760
the sum of the reversed number and the original number is 154, find the original number, if the ones digit in it is 2 less than the tens digit
==========================================================
Explanation:
For the original two digit number, we have:
x = tens digity = units digit (aka ones digit)The original number is 10x+y. For instance, if x = 5 and y = 7, then we have 10x+y = 10*5+7 = 57.
This original number 10x+y swaps its digits to 10y+x. Then we add up the two expressions:
(original)+(swapped) = (10x+y)+(10y+x) = 11x+11y = 11(x+y)
This sum is stated to be 154
11(x+y) = 154
x+y = 154/11
x+y = 14
Then we're told that "the ones digit in it (the original number) is 2 less than the tens digit". Meaning that y = x-2 is another equation we can use. We'll plug this into the previous equation we found and solve for x
x+y = 14
x+x-2 = 14 ... replace y with x-2
2x-2 = 14
2x = 14+2
2x = 16
x = 16/2
x = 8 is the tens digit of the original number
y = x-2
y = 8-2
y = 6 is the units digit of the original number
The original number is 10x+y = 10*8+6 = 86
As a check, adding 86+68 = 154 which confirms our answer.
-1/15+-(3/5). How do you do this problem? Is it positive or negative?
Answer:
it's a negative answer Because there's one negative and one positive and if there are your two negatives that makes a positive and it's also the same with two positives
what is the slope intercept form
Answer:
[tex]y = \frac{4}{3} x - 4[/tex]
Hope it helped:)
Is the sequence arithmetic or geometric?
1, 2, 4, 8, 16,...
Answer:
Step-by-step explanation:
Geometric sequence.
Each term is obtained by multiplying 2. So, it is geometric sequence
2 = 1*2
4 = 2*2
8 = 4 *2
16 = 8 *2
PLEASE HELP WITH ALL
Answer:
For 9., the answer is x = -6
For 10., the answer is x = 6
For 11., the answer is x = -7
For 12., the answer is x = -6
write equations for each of the following graphs!
NO LINKS!!
9514 1404 393
Answer:
y = √(x+4) -2y = -(x -2)^2 -3Step-by-step explanation:
The translation of a parent function f(x) by (h, k) gives the function ...
g(x) = f(x -h) +k . . . . . h units right, k units up
Reflection across the x-axis multiplies the function value by -1.
__
a) The square root function is translated 4 left and 2 down. The equation of the graph is ...
y = √(x +4) -2
__
b) The equation of the parabola is reflected across the x-axis and translated right 2 and down 3.
y = -(x -2)^2 -3
PLEASE HELP clear up my confusion! Carrie took out a $12,000 personal loan to pay off her credit cards. She will not make a payment for 2 years and there is a 5% interest rate. How much will be owed in 2 years with monthly compounding?
I know this is relatively simple but the variable that she doesn't pay for 2 years is confusing me on how to set up the problem.
Answer:
$13,259.30
Step-by-step explanation:
The account balance with compound interest is ...
A = P(1 +r/n)^(nt)
A = $12,000(1 +0.05/12)^(12·2) ≈ $13,259.30 . . . . will be owed in 2 years
__
It's a straight compound interest problem. "No payments for 2 years" means the account continues to accrue compounded interest. It's basically a future-value problem, as opposed to an amortization or annuity problem where payments are involved.
possible answers: 3.456 x 10^3, 3.456 x 10^4, 3.456 x 10^5, 3.456 x 10^6
Answer:
last option is correct
............................
Kevin’s car can go 315 miles on one tank of gas. He used just under 40% of a full tank of gas to get him to a sporting event Traveling at an average speed of 60 miles per hour about how long did it take him to get there how did you decide
Answer:
2 hours 6 minutes
Step-by-step explanation:
40% of 315 is 126 [miles]
126 miles = 126 minutes, since he went at 60 miles per hour [one mile per minute]
126 minutes = 2 hours 6 minutes
So, 2 hours 6 minutes, or 126 minutes.
Slope from graph Khan Academy. PLEASE HELP ME!! IM HAVING SO MUCH TROUBLE
Answer:
Step-by-step explanation:
you first have to find two points that pass directly through the graph in this case we have (1,4) and (4,-4)
Now we use y2-y1 divided by x2-x1
in this case -4 - 4 (y2-y1) and 4 - 1 (x2-x1)
giving you -8 and 3
so try putting -8/3 as your answer this is what i would put sorry if it is incorrect
(3a - b)(a + b) - (3a - b)²
I like to have some leftovers.
I've discovered that the
proper ratio of turkey to
people is 12 pounds of turkey
for every'8 people. If 36
people are going to be
eating, how many pounds of
turkey do I need to buy?
How do you know if a table of values shows a function?
Work out m and c for the line: 2 y + 6 x = − 4
Answer:
mn
Step-by-step explanation:
How to solve f(x)-2x+1 find f(h+2)
Step-by-step explanation:
In pictures answered .
Write the product using exponents.
(-0.9)×(-0.9)×(-0.9)
Answer:
-0.9³
Step-by-step explanation:
that's the answer hope it helps
Answer:
(-0.9)×(-0.9)×(-0.9)
(-0.9)^3
What is the value of the expression 1/7 divided by five
Answer:
1/35
Step-by-step explanation:
plz mark brainllest
Answer:
1/7 divided by 5 =
1/35
in decimal form, its:
0.0285714...
Give 3 points that all lie on a line with a slope of 4. Explain your process.
Answer:
Step-by-step explanation:
The simplest line you can draw with a slope of 4 is y = 4x. Use The equation to make a table
y = 4x
x 1 3 5
y 4 12 20
So I just multiplied 4 by 3 and 5 red graph below shows the three points on the line. (1,4),(3,12),(5,20)
What about a more complex line.
y = 4x + 5
x 1 3 5
y 9 17 25
The green line show (1,9),(3,17),(5,25)
Nikki descended 20.4 meters from the ocean surface. She then ascended 13.6 meters.
How far from the ocean surface was she?
