Let S be the sum of the integers 25-210:
S = 25 + 26 + 27 + … + 208 + 209 + 210
Let S₃, S₄, and S₁₂ denote the sums of the integers in S that are multiples of 3, 4, or 12, respectively. We'll also count how many terms each sum involves; it'll be useful later.
S₃ = 27 + 30 + 33 + … + 204 + 207 + 210 … … … (62 terms)
S₃ = 3 (9 + 10 + 11 + … + 68 + 69 + 70)
S₄ = 28 + 32 + 36 + … + 200 + 204 + 208 … … … (46 terms)
S₄ = 4 (7 + 8 + 9 + … + 50 + 51 + 52)
S₁₂ = 36 + 48 + 60 + … + 180 + 192 + 204 … … … (15 terms)
S₁₂ = 12 (3 + 4 + 5 + … + 15 + 16 + 17)
Let's look at S₃ :
S₃ = 3 (9 + 10 + 11 + … + 68 + 69 + 70)
By reversing the order of the sum, we get
S₃* = 3 (70 + 69 + 68 + … + 11 + 10 + 9)
Of course S₃ = S₃*, I'm just calling it something else temporarily. Notice that every term in the same position of either sum adds up the same number.
9 + 70 = 79
10 + 69 = 79
11 + 68 = 79
and so on. Then
S₃ + S₃* = 3 (79 + 79 + 79 + … + 79 + 79 + 79)
or
2S₃ = 3 × 62 × 79 ==> S₃ = 7,347
We can compute the other two sums in the same way.
S₄ = 4 (7 + 8 + 9 + … + 50 + 51 + 52)
S₄* = 4 (52 + 51 + 50 + … + 9 + 8 + 7)
==> 2S₄ = 4 × 46 × 59 ==> S₄ = 5,428
S₁₂ = 12 (3 + 4 + 5 + … + 15 + 16 + 17)
S₁₂* = 12 (17 + 16 +15 + … + 5 + 4 + 3)
==> 2S₁₂ = 12 × 15 × 20 ==> S₁₂ = 1,800
Then the sum you want is
S₃ + S₄ - S₁₂ = 10,975
We subtract S₁₂ because each of its terms is counted twice (once in S₃ and again in S₄).
The Jones family was one of the first to come to the U.S. They had 9 children. Assuming that the
probability of a child being a girl is .5, find the probability that the Jones family had:
at least 7 girls?
at most 8 girls?
Answer:
At least 7 girls: 0.5^7 OR 0.0078125
At least 8 girls: 0.5^8 OR 0.00390625
Let 0° < a < 90°
Given: cos a=7/25
Find: sin a and cot a
The solutions are:
sin(a) = 24/25
tan(a) = 24/7
For a given point (x, y) and an angle "a" measured counterclockwise from the positive x-axis to a ray that connects the origin with our point, we can think on the situation as a triangle rectangle.
Where the ray is the hypotenuse, the x-component is the adjacent cathetus, and the y-component is the opposite cathetus.
So we have:
x = adjacent cathetus
y = opposite cathetus
h = hypotenuse = √(x^2 + y^2)
Then the trigonometric relations become:
cos(a) = x/√(x^2 + y^2)
sin(a) = y/√(x^2 + y^2)
tan(a) = y/x
Now, we know that we have:
cos(a) = 7/25
then we can see that:
x = 7
and
h = 25 = √(7^2 + y^2)
We can solve the above equation for y:
25 = √(7^2 + y^2)
25 = √(49 + y^2)
25^2 = 49 + y^2
625 - 49 = y^2
√576 = y = 24
Then we have:
x = 7
y = 24
h = 25
Now we can return to our known trigonometric relations and get:
sin(a) = y/√(x^2 + y^2) = 24/25
tan(a) = y/x = 24/7
If you want to learn more about trigonometry, you can read:
https://brainly.com/question/14746686
Một người gửi tiết kiệm 100 triệu đồng theo kỳ hạn gửi là 1 năm. Sau 5 năm người người đó nhận được số tiền là 153,86 triệu đồng. Hỏi lãi suất người này gửi là bao nhiêu (lãi kép hàng năm):
Answer:
no
Step-by-step explanation:
Help with Pythagorean theorem
Answer:
45
Step-by-step explanation:
(27*27)+(36*36) = 2025
square root 2025 and you get 45
to check 45*45=2025
Nishi invests £3400 at 5% interest per year. Work out how much she will have altogether after: 3 years
Answer:
£510
Step-by-step explanation:
5% of 3400 is 170. SInce she gets that much money every year, after 3 years she'll have 3*170 = £510.
Answer: Simple Interest = PRT
P= 3400
R = 5% = 5/100
T = 3
=> 3400 × 5/100 × 3 = 510
=》3400 + 510 = 3910
Step-by-step explanation:
Answer please struggling
Answer:
x ≈ 28.2
Step-by-step explanation:
Δ CAB ≅ Δ CDE then corresponding sides are in proportion, that is
[tex]\frac{CA}{CD}[/tex] = [tex]\frac{CB}{CE}[/tex] , substitute values
[tex]\frac{14+x}{x}[/tex] = [tex]\frac{18.7+9.3}{18.7}[/tex] = [tex]\frac{28}{18.7}[/tex] ( cross- multiply )
28x = 18.7(14 + x) ← distribute
28x = 261.8 + 18.7x ( subtract 18.7x from both sides )
9.3x = 261.8 ( divide both sides by 9.3 )
x ≈ 28.2 (to the nearest tenth )
simplify 4x^2-3xy+2xy+9x^2
Answer:
13x^2 - xy
Step-by-step explanation:
4x^2-3xy+2xy+9x^2
=13x^2-xy
I NEED HELP JANSJEHEHSHSBSBSBSH
Answer:
The answer is a translation
Step-by-step explanation:
In Math, translation is the displacement of a shape or object from one place to another.
Since the picture shows that the shape moved from one place to the next while remaining the same size, it is translation.
If the odds against Deborahs winning first prize in a chess tournament are 1 to 11 what is the probability of the event that she will win first prize
Answer: [tex]\dfrac{11}{12}[/tex]
Step-by-step explanation:
Given
The odds against winning in a chess tournament are 1 to 11.
Odds is defined as the ratio of the probability of occurrence to the non-occurrence of event.
[tex]\therefore \text{Probability that event will occur is P'=}\dfrac{1}{1+11}\\\\\Rightarrow P'=\dfrac{1}{12}[/tex]
Probability of non-occurrence i.e. she wins the first prize is
[tex]\Rightarrow P=1-\dfrac{1}{12}\\\\\Rightarrow P=\dfrac{11}{12}[/tex]
Find the length of UV. v(2,-1) u(1,-9)
Answer: [tex]\sqrt{65}[/tex]
Step-by-step explanation:
Concept:
Here, we need to know the concept of the distance formula.
The distance formula is the formula, which is used to find the distance between any two points.
If you are still confused, please refer to the attachment below for a clear version of the formula.
Solve:
Find the length of UV, where:
U (1, -9) V (2, -1)[tex]Distance = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
[tex]Distance = \sqrt{(1-2)^2+(-9+1)^2}[/tex]
[tex]Distance = \sqrt{(-1)^2+(-8)^2}[/tex]
[tex]Distance = \sqrt{1+64}[/tex]
[tex]Distance = \sqrt{65}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
what is the value of the smallest of five consecutive integers if the least minus twice the greates equals -3
A. -9
B. -5
C. -3
D. 5
Answer: (b)
Step-by-step explanation:
Given
There are five consecutive integers and the least minus twice the greatest equals to -3
Suppose [tex]x,x+1,x+2,x+3,x+4[/tex] are the five consecutive integers
According to the question
[tex]\Rightarrow x-2(x+4)=-3\\\Rightarrow x-2x-8=-3\\\Rightarrow -x=8-3\\\Rightarrow x=-5[/tex]
option (b) is correct.
salini's mother's present age is 8 times salini's present age five years from now salini's age will be 1/4th of of his mother's present age what are their present ages
Answer:
salini -= 5 years
mother = 40years
Step-by-step explanation:
[tex]salini = x \\ mother =8x \\ x + 5 = 1 \: of \: 8x \\ x + 5 = 2x \\ 5 = x [/tex]
[tex]5 = x[/tex]
then you equate the values of x
One model of Earth's population growth is P(t)= 64/(1+11e^0.8t)
where t is
measured in years since 1990, and P is measured in years since 1990, and Pis measured in billions of people. Which of the following statements are true? Check all that apply.
Answer: C and D
Step-by-step explanation:
Using the logistic equation, it is found that options C and D are correct.
The logistic equation for population growth is given by:
[tex]P(t) = \frac{K}{1 + Ae^{-kt}}[/tex]
[tex]A = \frac{K - P(0)}{P(0)}[/tex]
In which:
K is the carrying capacity. P(0) is the initial value. k is the growth rate, as a decimal. The population grows exponentially for a while, but as it gets closer to the carrying capacity, the growth slows down.For this problem, the equation is:
[tex]P(t) = \frac{64}{1 + 11e^{-0.08t}}[/tex]
Which means that:
The carrying capacity is of 64 billion people, as [tex]K = 64[/tex].The growth rate is of 8% per year, but it is not steady.The initial population, in millions of people, is of [tex]P(0) = \frac{64}{1 + 11} = 5.3[/tex].Hence, options C and D are correct.
To learn more about the logistic equation, you can check https://brainly.com/question/25697660
Is 392 a perfect cube? If not, find the smallest natural number by which
392 must be multiplied so that the product will be a perfect cube.
Answer:
no its not a perfect cube, we need to multiply a seven to make it perfect cube
Step-by-step explanation:
answer from GAUTHMATH
Need answers please
Answer:
C
Step-by-step explanation:
you are adding
-3√18=
-27
√16=4
-27+4=-23
Phythagorean theorem help me plsss
Answer:
5
Step-by-step explanation:
We know that a^2+b^2=c^2 from the Pythagorean theorem where a and b are the legs and c is the hypotenuse
4^2+3^2 = c^2
16+9 = c^2
25 = c^2
Taking the square root of each side
sqrt(25) = sqrt(c^2)
5 =c
Answer:
d = 5 m
Step-by-step explanation:
The diagonal divides the rectangle into 2 right triangles with legs 3 and 4 and hypotenuse h
Using Pythagoras' identity in one of the right triangles, then
d² = 3² + 4² = 9 + 16 = 25 ( take the square root of both sides )
d = [tex]\sqrt{25}[/tex] = 5
When leaving a town, a car accelerates from 30 kmh-1 to 60 kmh-1 in 5 s. Assuming the
acceleration is constant, find the distance travelled in this time.
A. 6 m
B. 62.5 m
C. 41.7 m
D. 20.8 m
Answer:
B .62.5 m
Step-by-step explanation:
convert Kmh-1 in to ms-1
30 Kmh-1 = (30×1000) ÷3600 = 8.3ms-1
60 Kmh-1 = (60×1000) ÷3600 = 16.6 ms-1
acceleration = (16.6 - 8.3) ÷ 5 = 1.66 ms-2
V^2 = U^2 +2aS
(16.6)^2 = (8.3)^2 + 2×1.66 ×S
S = 62.5 m
If the graph of f(x) = x^2, how will the graph be affected if it is changed to f(x) = 3r^2?
Answer:
the graph curve will goes above f(x)=x^2.f(x)=3*x^2 curve will also give higher value same value of x .
Step-by-step explanation:
factorise. X^5+X^4+1.
Answer:
Answer
5.0/5
2
poojamaurya21
Virtuoso
133 answers
14.9K people helped
Answer:
): "x2" was replaced by "x^2". 4 more similar replacement(s).
Step by step solution :
Step 1 :
Step 2 :
Pulling out like terms :
2.1 Pull out like factors :
x6 - x5 + x4 - x3 + x2 - x =
x • (x5 - x4 + x3 - x2 + x - 1)
2.2 Factor x5 - x4 + x3 - x2 + x - 1
Try to factor this 6-term polynomial into (2-term) • (3-term)
Begin by splitting the 6-term into two 3-term polynomials:
-x2 + x - 1 and x5 - x4 + x3
Next simplify each 3-term polynomial by pulling out like terms:
-1 • (x2 - x + 1) and x3 • (x2 - x + 1)
Note that the two simplified polynomials have x2 - x + 1 in common
Now adding the two simplified polynomials we get
(x3 - 1) • (x2 - x + 1)
Can someone please check my answers and help/explain #4
Answer:
(0,8)
Step-by-step explanation:
To get the y-intercept put all x values equal to 0 (because we want to see where y lies on the graph when x is zero.
y=2^0+3
y=2^3
y=8
The y-intercept is (0,8)
As for the rest of the answers.....
1)A) 2[tex]\sqrt{12}[/tex]-5[tex]\sqrt{3}[/tex] = 4[tex]\sqrt{3}[/tex]-5[tex]\sqrt{3}[/tex] so the final answer is -[tex]\sqrt{3}[/tex]
1B) the problem is (3-[tex]\sqrt{7}[/tex])(3+[tex]\sqrt{7}[/tex]) a quick formula to help you solve this type of problem is (a-b)(a+b)= a^2-b^2 so you transform the question to 3^2-[tex]\sqrt{7}[/tex] ^2
to get 9-7 which is 3 therefore the final answer is 3
2) when you transform a function from y=x^2-3 to y=x^2+1 the graph moves 4 units up. So the final answer is b)4 units up
Question 3a and 3b are correct
CAN YALL HELP ME PLSSS NOW I NEED IT PLS IM BEGGING ITS FOR PYTHAGOREAN THEOREM
Answer:
The answer is 97!
Step-by-step explanation:
The formula for Pythagorean Theorem is a^2+b^2=c^2. (65)^2 + (72)^2 is 9409, which has a square root of 97. Hope this helped! :)
pls help me anyone! pls im so stuck here
Answer:
The third option, (-30,10)
Step-by-step explanation:
A proportional relationship is when you multiply one number in the coordinate/pair to get the other number, and that number that you multiply with is consistent. For example, in (60,-20) you multiply the x coordinate 60 by -1/3 to get 20. Next, you have to check if the other answer choices follow the same pattern. For (-30,10), you multiply -30 by -1/3 and get 10! So it works.
SEE QUESTION IN IMAGE
Answer:
b) 16.8 gStep-by-step explanation:
The modal group is 10-20, as it has the greatest frequency of 27.
Estimated mode is calculated by formula:
EM = L + (f(m) - f(m-1))/(f(m) - f(m-1) + f(m) - (f(m+1))*w,where
L- lower class boundary of the modal group =10 fm-1 - frequency of the group before the modal group = 10 fm - frequency of the modal group =27 fm+1 - frequency of the group after the modal group = 19 w - group width = 10Substitute values to get:
EM = 10 + (27 - 10)/(27 - 10 + 27 - 19)*10 = 16.8[tex]\sqrt{x^2 +7x+1} =2x+1[/tex]
X
45°
454
Find the value of x.
A.
B.
3.2
2
C. 3√2
D. 33
Save and Fyit
Give an example of a trinomial with a GCF of 6a.
9514 1404 393
Answer:
18a +42ax +24az
Step-by-step explanation:
6a(3 +7x +4z) = 18a +42ax +24az
__
Additional comment
In order for 6a to be the GCF, the terms inside parentheses cannot have any common factors.
please help asap!!! i dont understand it
Answer:
a
Step-by-step explanation:
A perpendicular bisector, intersects a line at its mid point and is perpendicular to it.
Calculate slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 7, 1) and (x₂, y₂ ) = (9, 13)
m = [tex]\frac{13-1}{9-(-7)}[/tex] = [tex]\frac{12}{9+7}[/tex] = [tex]\frac{12}{16}[/tex] = [tex]\frac{3}{4}[/tex]
Given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{\frac{3}{4} }[/tex] = - [tex]\frac{4}{3}[/tex] ← slope of perpendicular bisector
Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
([tex]\frac{x_{1}+x_{2} }{2}[/tex], [tex]\frac{y_{1}+y_{2} }{2}[/tex] )
using (x₁, y₁ ) = (- 7, 1) and (x₂, y₂ ) = (9, 13) , then
midpoint = ( [tex]\frac{-7+9}{2}[/tex], [tex]\frac{1+13}{2}[/tex] ) = ( [tex]\frac{2}{2}[/tex], [tex]\frac{14}{2}[/tex] ) = (1, 7 )
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Here m = - [tex]\frac{4}{3}[/tex] , then
y = - [tex]\frac{4}{3}[/tex] x + c ← is the partial equation
To find c substitute the midpoint (1, 7) into the partial equation
7 = - [tex]\frac{4}{3}[/tex] + c ⇒ c = [tex]\frac{21}{3}[/tex] + [tex]\frac{4}{3}[/tex] = [tex]\frac{25}{3}[/tex]
y = - [tex]\frac{4}{3}[/tex] x + [tex]\frac{25}{3}[/tex] ← equation of perpendicular bisector
Glven: 3x < 9.
Choose the solution set.
Answer:
x<3
Step-by-step explanation:
We divide equation by 3. So we get x<3. You can think of it with logic, too.
Answer:
x<3
Step-by-step explanation:
3x< 9
Divide each side by 3
3x/3 <9/3
x < 3
There is an open circle at 3 and a line going to the left
Please find the missing number for this surface answer! I will mark brainiest if correct! 1.
Answer:
2
Step-by-step explanation:
[tex](2 * 6x) + (2 * 7x) + (2 * (6 * 7)) = 136[/tex]
[tex]12x + 14x + (2 * 42) = 136[/tex]
[tex]12x + 14x + 84 = 136[/tex]
[tex]12x + 14x = 136 - 84[/tex]
[tex]26x = 52[/tex]
[tex]x = 2[/tex]
Question 3 of 10
Is ASAM-ADEL? If so, identify the similarity postulate or theorem that
applies.
A. Similar - AA
B. Similar - SSS
C. Similar - SAS
D. Cannot be determined
Answer:
B. SAS
[tex] \frac{sm}{dl} = \frac{27}{9} = 3 \\ \frac{am}{el} = \frac{15}{5} = 3 \\ angle \: m= angle \: l[/tex]
Brainliest please~
By SAS similarity, ΔASM is similar to ΔDLE. Thus, option C is correct.
What is SAS?The SAS theorem is also known as the side-angle-side theorem, also known as a statement in Euclidean geometry that states two triangles are congruent if their corresponding sides are both the same length and their included angles are both of the same measures.
According to the given figure,
SM=27
DL=9
MA=15
LE=5,
∠M=∠L=31°
It is required to determine the similarity postulate that applies. Therefore,
From the ΔASM and ΔDLE, we have
[tex]\frac{SM}{DL}=\frac{27}{9} \\=3\\\frac{ML}{LE}=\frac{15}{5}\\ =3\\ Hence, \frac{SM}{DL}= \frac{ML}{LE}[/tex]
∠M=∠L=31°
Hence, by SAS similarity, ΔASM is similar to ΔDLE.
Therefore, it can be concluded that option C is correct.
Learn more about the SAS theorem here:
https://brainly.com/question/31243316
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