Given:
Summation problem is:
[tex]\sum\limits_{n=1}^{35}n(n^2+1)[/tex]
To find:
The properties or formulas must be used to evaluate the given summation.
Solution:
We have,
[tex]\sum\limits_{n=1}^{35}n(n^2+1)[/tex]
In can be rewritten as:
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=\sum\limits_{n=1}^{35}(n^3+n)[/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=\sum\limits_{n=1}^{35}n^3+\sum\limits_{n=1}^{35}n[/tex] [tex]\left[\because \sum\limits_{i=1}^n(a_i\pm b_i)=\sum\limits_{i=1}^na_i\pm \sum\limits_{i=1}^nb_i\right][/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=\left[\dfrac{35(35+1)}{2}\right]^2+\dfrac{35(35+1)}{2}[/tex] [tex]\left[\because \sum\limits_{i=1}^ni=\dfrac{n(n+1)}{2},\sum\limits_{i=1}^ni^3=\left[\dfrac{n(n+1)}{2}\right]^2\right][/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=\left[\dfrac{35(36)}{2}\right]^2+\dfrac{35(36)}{2}[/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=\left[630\right]^2+630[/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=396900+630[/tex]
[tex]\sum\limits_{n=1}^{35}n(n^2+1)=397530[/tex]
Therefore, the properties and formula used are:
[tex]\sum\limits_{i=1}^n(a_i\pm b_i)=\sum\limits_{i=1}^na_i\pm \sum\limits_{i=1}^nb_i,\\\sum\limits_{i=1}^ni=\dfrac{n(n+1)}{2},\\\sum\limits_{i=1}^ni^3=\left[\dfrac{n(n+1)}{2}\right]^2[/tex]
Answer: see attachment below
Step-by-step explanation:
How to solve a problem.
a) [tex]\log \left(\dfrac{A^3B}{C} \right) = 3\log A + \log B - \log C[/tex]
b) [tex]\log \left(\dfrac{\sqrt{A}}{B^2} \right) = \frac{1}{2}\log A - 2\log B[/tex]
A tortoise and hare are competing in a 1000 meter race. The function f and g are such that f(t) represents the tortoise's distance from the finish line (in meters) t seconds after the start of the race and g(t) represents the hare's distance from the finish line (in meters) t seconds after the start of the race. If we want to compute the number of seconds needed for the hare to finish the race, which procedures should be used?
Answer:
[tex]g(t) = 0[/tex]
Step-by-step explanation:
Given
[tex]d = 1000m[/tex]
[tex]f(t) \to[/tex] Tortoise
[tex]g(t) \to[/tex] Hare
Required
Time needed by the Hare to complete the race
The function [tex]g(t)[/tex] implies that the Hare is t seconds from completing the race.
When g(t) = 0, it means that the Hare has completed the race
So, to calculate the time needed by the Hare;
We simply equate g(t) to 0, then solve for t
how can i solve the following
4(5x-2) = 2(9x + 3
Hank can see the top of a tree in a mirror that is placed 475 cm from the tree when he stands 190 cm from the mirror. What is the height of the tree?
Answer:
The height of the tree is 430cm
Step-by-step explanation:
Given
The attached illustration
Required
The height of the tree
Let:
[tex]h \to[/tex] Hank's height
[tex]H \to[/tex] Tree height
[tex]d \to[/tex] distance between Hank and Mirror
[tex]D \to[/tex] distance between tree and Mirror
From the question, we have:
[tex]\angle 1 = \angle 2[/tex]
This means that:
[tex]\frac{H}{D} = \frac{h}{d}[/tex]
Make H the subject
[tex]H = \frac{h}{d} * D[/tex]
So, we have:
[tex]H = \frac{172cm}{190cm} * 475cm[/tex]
[tex]H = \frac{172* 475cm}{190}[/tex]
[tex]H = \frac{81700cm}{190}[/tex]
[tex]H = 430 cm[/tex]
which of the following graph is neither even or odd
Answer:
f(x)=2x³+4x²+5
Step-by-step explanation:
1) the rule of the even function is: f(-x)=f(x)
the rule of the odd function is: f(-x)=-f(x);
2) the function f(x)=2x³+4x²+5 is neither even or odd according to the rules above:
f(-x)= -2x³+4x²+5= -(2x³-4x²-5).
Put the following equation of a line into slope-intercept form, simplifying all
fractions.
9x + 3y = -6
Answer:
y = -3x-2
Step-by-step explanation:
3y = -9x-6. y = (-9x-6)/3. y = (-9x/3)+(-6/3). y = -3x-2
A scientist is studying the growth of a particular species of plant. He writes the following equation to show the height of the plant f(n), in cm, after n days:
f(n) = 12(1.03)n
What is the average rate of change of the function f(n) from n = 3 to n = 10, and what does it represent? (4 points)
Answer:
The average of rate was of 0.4314 cm/day, which means that on average, during this interval, the plant grew by 0.4314 cm each day.
Step-by-step explanation:
Average rate of change:
The average rate of change of a function f(x), from x = a to x = b, is given by:
[tex]A = \frac{f(b)-f(a)}{b-a}[/tex]
In this question:
[tex]f(x) = 12(1.03)^n[/tex]
From n = 3 to n = 10
[tex]a = 3, b = 10[/tex]
[tex]f(b) = f(10) = 12(1.03)^{10} = 16.13[/tex]
[tex]f(a) = f(3) = 12(1.03)^{3} = 13.11[/tex]
Average rate of change:
[tex]A = \frac{f(b)-f(a)}{b-a} = \frac{16.13-13.11}{10-3} = 0.4314[/tex]
The average of rate was of 0.4314 cm/day, which means that on average, during this interval, the plant grew by 0.4314 cm each day.
WILL MARK YOU IF YOU CAN HELP plz help me understand this to or add one of my socials to help me with geometry:(
Insta:aesthetictoxicbabe
Snap:Toxicangelbby
Answer:
17x + 2 = 16x+6
17x-16x= 6-2
x=4
Answer:
5.21212121
Step-by-step explanation:
The first thing to keep in mind is that a line is equal to 180. The first thing you sould start off by doing is write 17x+2+16x+6. Then to make it easier combine the like terms. Both combine 17x + 16x to get 33x, and 6 and 2 to get 8. Now the problem should be 33x + 8 = 180. Next we will take away 8 from both sides. This should eliminate 8 for us and give us 33x=180 as our next problem. Finally you will do the opposite of multiplying which is diving. 33 divided by itself gives us 0. So we will divide 172 by 33 to get 5.21212121.
YOU MAY PLUG THIS NUMBER BACK IN TO CHECK YOUR ANSWER.
the question is in the attachment. i know its not A because these angles are not opposite of each other.
Answer:
hi, it's b because a+b is 180°
Evaluate the work below
Answer:
[tex]3 \sqrt{2} [/tex]
Step-by-step explanation:
Given
[tex] \frac{ \sin(60) \sec(60) \cot(30) }{ \cos( 45) ) } [/tex]
Using the unit circle, we know that
[tex] \sin(60) = \frac{ \sqrt{3} }{2} [/tex]
[tex] \cos(45) = \frac{ \sqrt{2} }{2} [/tex]
We can find sec and cot by using reciprocal identies.
[tex] \sec(x) = \frac{1}{ \cos(x) } [/tex]
[tex] \cos(60) = \frac{1}{2} [/tex]
So
[tex] \sec(60) = 2[/tex]
[tex] cot(x) = \frac{1}{ \tan(x) } [/tex]
[tex] \tan(30) = \frac{1}{ \sqrt{3} } [/tex]
so
[tex] \cot(30) = \sqrt{3} [/tex]
Plug in the values.
[tex] \frac{ \frac{ \sqrt{3} }{2} \times 2 \times \sqrt{3} }{ \frac{ \sqrt{2} }{2} } [/tex]
[tex] \frac{ \frac{ \sqrt{3} }{2} \times \frac{ \sqrt{12} }{1} }{ \frac{ \sqrt{2} }{2} } [/tex]
[tex] \frac{3}{ \frac{ \sqrt{2} }{2} } [/tex]
[tex]3 \sqrt{2} [/tex]
Find the missing length indicated. Please help!!!
5.) $28,000 invested at a rate of 4% compounded annually
a.) Write a compound interest function to model the situation.
Answer:
ok so we know the amount we know the interest amount but we do know know the period of time so the equation would look like this
28000(0.4)T=I
T=Time
I=interest
Find the area of the polygon.
is this right?
ERGENT
Answer:
The answer is in the picture.
15cm²
Pls Help I'm really really stumped.
9514 1404 393
Answer:
see attached
Step-by-step explanation:
The graph shown with your problem statement does not pass the vertical line test, so is not the graph of a function. (A vertical line intersects the graph in more than one place.) This graph can be rejected immediately.
I find it convenient to look at the domain statements (x≤1 and x>1). This tells you that you can separately evaluate the graph to the left of x=1 and to the right of x=1. Each of those portions of the graph must match the corresponding expression.
Here, it looks sufficient to identify the slope of the line in each domain. (We don't know what graphs you're selecting from, so we don't know what you will need to look at to differentiate the correct one from the rest.)
For x < 1, the coefficient of x is 3, so the line will have a steep upward slope.
For x > 1, the coefficient of x is -2, so the line will have a slightly less steep downward slope.
The value of z(1) is 3(1)-2=1, so (1, 1) is at the top end of the line forming the left side of the graph. The limit as x approaches 1 from the right is -2(1)+3=1. This means the two lines meet at the point (1, 1) and the graph has a generally ∧ shape. The attachment shows the graph you're looking for.
Find the area of the triangle to the nearest tenth . Pleaseeee help
Answer:
15.8 cm²
Step-by-step explanation:
Please see attached photo for diagram.
We'll begin by calculating the angle A. This can be obtained as follow:
B = 56°
C = 78°
A =?
A + B + C = 180 (Sum of the angle in triangle)
A + 56 + 78 = 180
A + 134 = 180
Collect like terms
A = 180 – 134
A = 46°
Next, we shall determine the value of b by using the sine rule. This can be obtained:
Side opposite angle C (c) = 7.2 cm
Angle C = 78°
Angle B = 56°
Side opposite angle B (b) =?
b/Sine B = c/sine C
b/Sine 56 = 7.2/Sine 78
Cross multiply
b × Sine 78 = 7.2 × Sine 56
Divide both side by Sine 78
b = 7.2 × Sine 56 / Sine 78
b = 6.1 cm
Finally, we shall determine the area of the triangle. This can be obtained as follow:
Side opposite angle C (c) = 7.2 cm
Side opposite angle B (b) = 6.1 cm
Angle A = 46°
Area (A) =?
A = ½bcSineA
A = ½ × 6.1 × 7.2 × Sine 46
A = 15.8 cm²
Therefore, the area of the triangle is 15.8 cm²
2x + 3y = 7 and x + y = 3? *
Answer:
1) 7/2 - 3/2y,
2) 3-y, y
Step-by-step explanation:
See Image below for steps:)
Can someone please help me with this math problem? thank you so much!
9514 1404 393
Answer:
(A) 6
Step-by-step explanation:
Divisors of 6 are ...
1, 2, 3, 6
The sum of all but the last is 1+2+3 = 6.
Then <<<6>>> = <<6>> = <6> = 6.
_____
Additional comment
Among integers less than 200, only 6 and 28 are solutions to <n> = n.
If two events are complementary, then we know that: Multiple Choice the sum of their probabilities is one. the joint probability of the two events is one. their intersection has a nonzero probability. they are independent events.
Answer:
The joint probability of the two events is one.
Step-by-step explanation:
Complementary events:
If two events are complimentary, these three following things are true:
They are dependent.
The intersection of them is zero.
The joint probability of the two events is one.
The last one is the correct choice.
3
3
X
14. Aşağıdaki cümlelerin hangisinde ikilemenin
oluşum yolu diğerlerinden farklıdır?
L
A) Çamurda bata çıka ilerlemeye çalıştık.
er-
B) Denizde irili ufaklı taşlar vardı.
C) Maaşımla iyi kötü geçinip gidiyoruz.
D) Eski püskü eşyaları kenara ayırdı.
es
?
Answer:
hygu
Step-by-step explanation:
yg9k
please answer both questions
Answer:
$24.63
Step-by-step explanation:
Surface area of a sphere = 4πr²
Surface area of the sphere
= 4π7²
= 196π
= 615.752... cm²
The total cost
= 0.04 × 615.752...
= $24.63 (rounded to the nearest hundredth)
What is the perimeter of the quadrilateral ABCD?
Given:
The figure of a quadrilateral ABCD.
To find:
The perimeter of the quadrilateral ABCD.
Solution:
In an isosceles triangle, the two sides and base angles are congruent.
In triangle ABD,
[tex]\angle DAB\cong \angle ABD[/tex] [Given]
[tex]\Delta ABD[/tex] is an isosceles triangle [Base angle property]
[tex]AD=BD[/tex] [By definition of isosceles triangles]
[tex]8=BD[/tex] ...(i)
In triangle BCD,
[tex]\angle BCD\cong \angle CDB\cong \angle CBD[/tex] [Given]
All interior angles of the triangle BCD are congruent, so the triangle BCD is an equilateral triangle and all sides of the triangle area equal.
[tex]BC=CD=BD[/tex]
[tex]BC=CD=8[/tex] [Using (i)] ...(ii)
Now, the perimeter of quadrilateral ABCD is:
[tex]Perimeter=AB+BC+CD+AD[/tex]
[tex]Perimeter=11+8+8+8[/tex]
[tex]Perimeter=35[/tex]
Therefore, the perimeter of the quadrilateral ABCD is 35 units.
What is the area of this tile?
HELPP PLEASEEE
12? im not sure
i think u need to multiply 6 and 2 to get your answer.
By what percent will a fraction increase if its numerator is decreased by 60% and its denominator is decreased by 20%?
9514 1404 393
Answer:
-50%
Step-by-step explanation:
Suppose the fraction is f = p/q before any changes.
Decreasing the numerator by 60% multiplies it by (1 -0.60) = 0.4.
Decreasing the denominator by 20% multiplies it by (1 -0.20) = 0.8.
Then the new fraction is ...
f' = p(0.4)/(q(0.8)) = (p/q)(0.4/0.8) = f(0.5) = f(1 -0.50) = f(1 +(-50%))
The increase in the value of the fraction is -50%.
i need help with this one plz! (marking BRAINLIST) :)
Answer:
x=22
Step-by-step explanation:
the triangle in which x is in is an equilateral triangle.
44 is an exterior angle to that triangle.
an exterior angle is always equal to tbe sum of the opposite 2 interior angles.
because that triangle is equilateral
x =44 ÷ 2
x=22
5. Solve: 9(36 – 2) - 2 =
I can’t figure this out
Answer:
The correct answer is:304
Step-by-step explanation:
Whoever answers correct first I will give BRAINLIEST!!
How many fives
are in 100?
Answer:
20
Step-by-step explanation:
100/5=20
¿ que pasa si no hubieran plantas ?
Answer:
entonces no habria oxigeno
Step-by-step explanation:
What is the 5 steps in Method of Elimination Steps to Solve Simultaneous Equations..
Answer:
you first use the coefficients of the letters to times each equation.
Then you either subtract or divide to eliminate the first letter and it coeeffient.
then whatever you get you will divide it with the one with no letter.
then you substitute the answer for the letter in any of the equation.
then you solve it.
I hope this is helpful.
I need help with this question
Answer:
56 degrees
Step-by-step explanation:
We know, the sum of the three angles of any triangle is equal to 180 degrees.
63 + 61 = 124
Now, 180 - 124 = 56 degrees
Hope this will help. Please give me brainliest.
find the value of x . PLEASE ANSWER QUICK ILL MARK BRAINLIEST
Answer:
x = 20
Step-by-step explanation:
90° - 76° = 14°
[tex](x-6)=14[/tex]
[tex](x+6)=+6[/tex] Add 6 to both sides
x = 20
Hence, x = 20
Answer: X=20
Right angles are 90 degrees and 90-76=14... X-6=14... add 6 to 14 and get 20. X=20. Don't quote me on it I haven't done this stuff in a year.