The binomial theorem says
[tex]\displaystyle (a + b)^n = \sum_{k=0}^n \binom nk a^{n-k} b^k \\\\(a+b)^n= \binom n0 a^n + \binom n1 a^{n-1}b + \cdots + \binom n{n-1} ab^{n-1} + \binom nn b^n[/tex]
where
[tex]\dbinom nk = \dfrac{n!}{k!(n-k)!}[/tex]
is the so-called binomial coefficient. The binomial coefficients follow a neat pattern called Pascal's triangle:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
where the n-th row, starting with n = 0, lists the coefficient of the k-th term in the sum (with 0 ≤ k ≤ n). For example,
[tex]\displaystyle n=0 \implies (a + b)^0 = \sum_{k=0}^0 \binom nk a^{0-k} b^k = \binom00 a^{0-0} b^0 = \underline{1}[/tex]
[tex]\displaystyle n=1 \implies (a + b)^1 = \sum_{k=0}^1 \binom 1k a^{1-k} b^k = \binom10 a^{1-0} b^0 + \binom11 a^{1-1}b^1 = \underline{1}a + \underline{1}b[/tex]
[tex]\displaystyle n=2 \implies (a + b)^2 = \sum_{k=0}^2 \binom 2k a^{2-k} b^k = \binom20a^{2-0}b^0+\binom21a^{2-1}b^1+\binom22a^{2-2}b^2 = \underline{1}a^2 + \underline{2}ab+\underline{1}b^2[/tex]
Each row starts and ends with 1, and every coefficient in between is obtained by adding the coefficients directly above and to the left. The next row, for instance, would be
1 1 + 4 4 + 6 6 + 4 4 + 1 1
or
1 5 10 10 5 1
which is to say,
[tex]\displaystyle (a + b)^5 = \underline{1}a^5+\underline{5}a^4b+\underline{10}a^3b^2+\underline{10}a^2b^3+\underline{5}ab^4+\underline{1}b^5[/tex]
In your case, we have (d - 4b)², so we take a, b, and n above with d, -4b, and 2, respectively:
[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k d^{2-k} (-4b)^k[/tex]
[tex]\displaystyle (d - 4b)^2 = \sum_{k=0}^2 \binom 2k (-4)^k d^{2-k} b^k[/tex]
[tex]\displaystyle (d - 4b)^2 = \binom20 (-4)^0 d^{2-0} b^0 + \binom21 (-4)^1 d^{2-1} b^1 + \binom22 (-4)^2 d^{2-2} b^2[/tex]
[tex]\displaystyle (d - 4b)^2 = 1\cdot1\cdot d^2\cdot1 + 2 \cdot (-4) d \cdot b + 1 \cdot 16 \cdot 1 \cdot b^2[/tex]
[tex]\boxed{(d - 4b)^2 = d^2 - 8bd + 16b^2}[/tex]
Solve for P 2(p-10)= 14
Hi,
2(p-10) = 14
2p - 20 = 14
2p - 20 + 20 = 14 + 20
2p = 34
2p/2 = 34/2
p = 17
✅(•‿•)
Find the
17th
term of the arithmetic sequence whose common difference is d=5 and whose first term is a1 = 1.
The [tex]17th[/tex] term of the arithmetic sequence is [tex]81[/tex]
An arithmetic sequence is a sequence of numbers produced by adding a fixed number, called the common difference, to a term in the sequence to get the next term of the sequence.
The formula for calculating the [tex]nth[/tex] term of a sequence is
[tex]a_n=a_1+(n-1)d[/tex]
where
[tex]a_n=\text{the nth term}\\a_1=\text{the first term}\\d=\text{the common difference}[/tex]
from the question
[tex]n=17\\d=5\\a_1=1[/tex]
from this information, we can calculate the [tex]17th[/tex] term of the sequence thus
[tex]a_n=a_1+(n-1)d\\a_{17}=1+(17-1)\times5\\=81[/tex]
so, [tex]a_n=81[/tex]
Learn more about arithmetic sequences here: https://brainly.com/question/16130064
help pl pls pls pls pls
Answer:
they are both a function
Step-by-step explanation:
because i did this back in 8th grade
How many weeks will Sanjay need to ride his bike to work in order to ride 150 kilometers?
Answer:
This is because he rides 37.5 k/m every week. Multiply that by four and you get 150
Answer:
Sanjay will need to ride his bike to work for 4 weeks in order to ride 150 kilometers.
CAN SOMEONE HELP ME PLEASE ASAP!?
Answer:
15
Step-by-step explanation:
7.5*2
Answer:
15 ft
Step-by-step explanation:
1 inch is 2 ft
remember 1 ft is 12 inches so,
1 inch is 24 inches
so 7.5 inch will be 24*7.5 = 180 inches
180 inch / 12 = 15 ft
What is the greatest common factor of 31 and 45?
The Answer is 15,good look!
Greatest Common Factor of 31 and 45 is :
(i) 1someone help me please
Answer:
8
Step-by-step explanation:
Basically insert 18 where the X is and go from there!
2/3 (18) - 4
2/3 × 18 = 12
12 - 4 = 8
Ben worked 5 hours at x dollars per hour. Julie worked 8 hours
and earned twice as much as Ben per hour. Phil worked 10
hours and earned 4 dollars an hour less than Julie. Choose the
expression that is correctly simplified and best represents
their total wages.
A. 31x - 4
B. 31x – 40
C. 41x - 10
D. 41x -40
Which event is important enough to include in a summary of this
story? Add the event to the chart.
D Event
Event
Event
Hester gives Ms.
Walker the letter
explaining her idea.
Ms. Walker agrees
to consider
Hester's idea.
?
Hester repeats a
*) phrase in her mind
over and over.
D
Hester packs up
her things when
the bell rings.
D
Hester returns to
her desk to read
her book.
Hester learns
about another way
to boost her grade.
When making a summary, it is important to read and understand the text which you are writing.
What is a Summary?This refers to the concise representation of the main points of a literary text without the use of bias.
Hence,w we can see that because the question is incomplete, you would need a general overview to understand it, so it would be best if you:
Read the given textUnderstand the plot and characterizationFollow the Rising Action, Climax, and Falling ActionList out the main pointsThen, join them in order of occurrence.Read more about summary here:
https://brainly.com/question/25605883
#SPJ2
Triangle XYZ is shown on the coordinate plane below:
Answer:
(4, -3)
Step-by-step explanation:
X is 4 units away from y=1, so 4 units further away is -3. The Y value of the point stays the same.
100 people planned to run to raise funds for a charity. Twenty percent of them ran 3 miles, two-fifth ran 6 miles, and the rest ran 9 miles. For every mile run, they raised $20. What was the total amount raised for the charity?
Answer:
$13,200
Step-by-step explanation:
20*3=60*20=1200
40*6=240*20=4800
40*9=360*20=7200
1200+4800+7200=13200
ANSWER:
360
Step-by-step explanation:
Compare 2/5 and 8/10
Answer:
2/5 is 4/10. 8/10 is 2 time greater than 2/5.
Step-by-step explanation:
4/10= 0.8, which can be written 8/10.
Make me brainlist.
If f(x) = 4x − 1 and g(x) = −3x + 2 (a) Find ℎ(x) = f(x) + g(x)
Answer:
x+1
Step-by-step explanation:
4x - 1 + (-3x) + 2 = x + 1
2(4w+7)=62
Whats the answer how do I do it?
[tex]2(4w + 7) = 62[/tex]
Divide both sides by 2
[tex] \frac{2(4w + 7)}{2} = \frac{62}{2} \\ [/tex]
[tex]4w + 7 = 31[/tex]
Subtract both sides 7
[tex]4w + 7 - 7 = 31 - 7[/tex]
[tex]4w = 24[/tex]
Divide both sides by 4
[tex] \frac{4w}{4} = \frac{24}{4} \\ [/tex]
[tex]w = 6[/tex]
Answer:
w=6
Step-by-step explanation:
Step 1 : Simplifying
2(4w + 7) = 62
Step 2 : Recorder the terms :
2(7 + 4w) = 62
(7 * 2 + 4w * 2) = 62
(14 + 8w) = 62
Solving-
14 + 8w = 62
Solving for the variable 'w'
Move all terms containing w to the left side, all other terms to the right.
Add '- 14' to each side of the equation.
14 + - 14 + 8w = 62 + - 14
Combine like terms: 14 + - 14 = 0
0 + 8w = 62 + - 14
Combine like terms: 62 + - 14 = 48
8w = 48
Divide each side by '8'
W = 6
Simplifying
W = 6
How many solutions exist for the system of equations in the graph?
CAN SOMEONE HELP ME PLEASE ASAP!?
Answer:
The airplane is 12 ft long.
Step-by-step explanation:
If one inch is 2 feet in real life, if there are six inches, that means there are 12 feet because it is a one to two ratio. :)
Answer:
12 ft
Step-by-step explanation:
what is the greatest common multiple of 9 and 27
Hi,
The greatest common multiple of 9 and 27 is 9
explanations:
9 x 1 = 9
9 x 3 = 27
✅;-)
Peyton knows that 1/8 means “1 divided by 8.” She uses this to find the decimal equivalent for 1/8
Answer:
Peyton can multiply 1 by 100, to make 100 over 8, and divide 100 by 8 which is 12.5 then dividing by 100 to get .125 which is equal to 1/8
Step-by-step explanation:
Answer: I am assuming the question is asking if Peyton got the decimal equivalent of 1/8 correct. Well, since the question is incomplete, I do not know what Peyton got, however, I can tell you that the decimal equivalent for 1/8 is 0.125
Solve for x
2x+5=33
Answer: x=14
hopes this helps
Answer: your anwser is 19
Step-by-step explanation:
Add 55 to both sides.
2x=33+5
2x=33+5
2 Simplify 33+533+5 to 3838.
2x=38
2x=38
3 Divide both sides by 22.
x=\frac{38}{2}
x=
2
38
4 Simplify
2
38
to 1919.
x=19
x=19
please mark brainliest
What is the slope of the line through (6,0) and (7,1)
[tex]\huge \bf༆ Answer ༄[/tex]
Let's find the slope of line passing through given points ~
[tex] \sf \dfrac{y2 - y}{x2 - x1} [/tex][tex] \sf \dfrac{1 - 0}{7 - 6} [/tex][tex] \sf \dfrac{1}{1} [/tex][tex] \sf1[/tex]Hence, slope of the line will be 1.
Answer:
1
Step-by-step explanation:
slope = (change in y)/(change in x)
slope = (1 - 0)/(7 - 6)
slope = 1/1 = 1
Answer: 1
For his statistics project, Martin recorded the weights of some apples.
Apples' weights (9)
+
70
80
90
100
110
120
What was the range of the apples' weights?
grams
Answer:
50 grams
Step-by-step explanation:
You simply subtract 70 from 120, and get 50.
Answer:
40 Grams!! i got it wrong and this was the answer
Step-by-step explanation:
Which account option is more convenient for frequent check or debit card withdrawals and purchases but gains no interest?
A. Checking
B. Certificate of deposit
C. Savings
D. Money market
what's 0.084507042 rounded
Answer:
i belive by rounding you get 1
Step-by-step explanation:
A
=
h
(
x
+
10
)
2
A
=
27
h
=
4
Work out the value of
x
.
Answer:
Step-by-step explanation:
A= h(x + 10)^2
27 = 4(x + 10)^2
(x + 10)^2 = 27/4
x + 10 = √27/2
x + 10 = 3√3/2
x = 3√3/2 - 10
= -7.40 to nearest hundredth.
Can someone plz help with 21?? Thanks!
Answer:
4) -11.3...
Step-by-step explanation:
distribute both sides to get (1/6)x - (4/3) = (4/15)x - (1/5)
move the variables to one side and the constants to the other: ((4/15) - (1/6))x = (-(4/3) + (1/5))
simplify to get (1/10)x = (-17/15) multiply everything by 10 to get x=-34/3 or -11.33333333333333333...
Answer:
option 4
Step-by-step explanation:
[tex]\frac{2}{3}[/tex] ([tex]\frac{1}{4}[/tex] x - 2) = [tex]\frac{1}{5}[/tex] ([tex]\frac{4}{3}[/tex] x - 1) ← distribute parenthesis on both sides
[tex]\frac{1}{6}[/tex] x - [tex]\frac{4}{3}[/tex] = [tex]\frac{4}{15}[/tex] x - [tex]\frac{1}{5}[/tex]
Multiply through by 30, the LCM of 6, 3, 15, 5
5x - 40 = 8x - 6 ( subtract 8x from both sides )
- 3x - 40 = - 6 ( add 40 to both sides )
- 3x = 34 ( divide both sides by - 3 )
x = - [tex]\frac{34}{3}[/tex] = - 11.333..
Please help due tmmr please help
Answer:
Leo´s response to the bully is a response of hurt and embarrassment. He seems sad and doesn´t even directly respond to the bullies ,¨several
students were huddled around Leo's desk... he had red rings around his eyes. He looked up at Hannah and said, "Why are people so mean?"
Explanation:
if Leo had posted something mean online the outcome would only have worsened. If it were him in the bullies shoes and the bully in his, the bully wouldve understood how these mean things make people feel.
Leo´s response to the bully is a response of hurt and embarrassment. He seems sad and doesn´t even directly respond to the bullies ,¨several
students were huddled around Leo's desk... he had red rings around his eyes. He looked up at Hannah and said, "Why are people so mean?"
Explanation:
if Leo had posted something mean online the outcome would only have worsened. If it were him in the bullies shoes and the bully in his, the bully wouldve understood how these mean things make people feel.
Leo´s response to the bully is a response of hurt and embarrassment. He seems sad and doesn´t even directly respond to the bullies ,¨several
students were huddled around Leo's desk... he had red rings around his eyes. He looked up at Hannah and said, "Why are people so mean?"
Explanation:
if Leo had posted something mean online the outcome would only have worsened. If it were him in the bullies shoes and the bully in his, the bully wouldve understood how these mean things make people feel.
Step-by-step explanation:
How is the graph of y = log (x) transformed to produce the graph of y = log (2 x) + 3?
It is stretched horizontally by a factor of 2 and translated up 3 units.
It is compressed horizontally by a factor of 2 and translated up 3 units.
It is stretched vertically by a factor of 2 and translated up 3 units.
It is compressed vertically by a factor of 2 and translated up 3 units.
Transformation involves changing the form of a function
The true statement is (a) it is stretched horizontally by a factor of 2 and translated up, 3 units.
The function is given as:
[tex]y = \log(x)[/tex]
Start by stretching the graph of [tex]y = \log(x)[/tex], horizontally by a factor of 2.
So, we have:
[tex]y = \log(2 \times x)[/tex]
[tex]y = \log(2x)[/tex]
Next, shift the function up by 3 units.
So, we have:
[tex]y = \log(2x) + 3[/tex]
Hence, the true statement is (a)
Read more about transformations at:
https://brainly.com/question/12619643
Answer:
its b on edge
Step-by-step explanation:
Explain what similarity means in terms of congruence and proportionality.
Answer: Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion . In other words, similar triangles are the same shape, but not necessarily the same size.
Step-by-step explanation: pls give brainlist
expand and simplify \left(4+\sqrt{2}\right)\left(6-\sqrt{2}\right) give your answer in the form of b +c \sqrt{2}
Answer:
you get 15\3 divide then you answer would be 5
Step-by-step explanation:
Find the length of side Y (this is from the geometry for enjoyment and challenge textbook) PLEASE ANSWER ASAP IT'S DUE TOMORROW!!!
Answer:
y= 3
Step-by-step explanation:
The triangle on the left is a right triangle, so we can use trig functions to solve for the missing side, y.
First, find the length of the side that both triangles are connected to.
tan30=x/9
Isolate the x variable and solve.
9(tan30)=x
x= 5.19615
Now use the Pythagorean Theorem to solve for the side length, y.
a^2+ b^2 = c^2
(5.19615)^2+ b^2= 6^2
c= 3