[tex]\sf 1^3+2^3\dots n^3=\dfrac{n^2(n+1)^2}{2}[/tex]
[tex]\\ \sf\longmapsto 1^3+2^3+\dots n^3=\left(\dfrac{n(n+1)}{2}\right)^2[/tex]
Solution:-Let
[tex]\\ \sf\longmapsto P(n)=1^3+2^3\dots n^3=\left(\dfrac{n(n+1)}{2}\right)^2[/tex]
For n=1
[tex]\\ \sf\longmapsto P(1)=\left(\dfrac{1(1+1)}{2}\right)^2[/tex]
[tex]\\ \sf\longmapsto P(1)=\left(\dfrac{1(2)}{2}\right)^2[/tex]
[tex]\\ \sf\longmapsto P(1)=\left(\dfrac{2}{2}\right)^2[/tex]
[tex]\\ \sf\longmapsto P(1)=(1)^2[/tex]
[tex]\\ \bf\longmapsto P(1)=1=1^3[/tex]
Let k be any positive integer.
[tex]\\ \sf\longmapsto P(k)= 1^3+2^3\dots k^3=\left(\dfrac{k(k+1)}{2}\right)^2[/tex]
We have to prove that p(k+1) is true.
consider
[tex]\sf 1^3+2^3\dots k^3+(k+1)^3[/tex]
[tex]\\ \sf\longmapsto \left(\dfrac{k(k+1)}{2}\right)^2+(k+1)^3[/tex]
[tex]\\ \sf\longmapsto \dfrac{k^2(k+1)^2}{4}+(k+1)^3[/tex]
[tex]\\ \sf\longmapsto \dfrac{k^2(k+1)^2+4(k+1)^3}{4}[/tex]
[tex]\\ \sf\longmapsto \dfrac{k+1)^2\left\{k^2+4k+4\right\}}{4}[/tex]
[tex]\\ \sf\longmapsto \dfrac{(k+1)^2(k+2)^2}{4}[/tex]
[tex]\\ \sf\longmapsto \dfrac{(k+1)^2(k+1+1)^2}{4}[/tex]
[tex]\\ \sf\longmapsto \left(\dfrac{(k+1)(k+1+1)}{2}\right)^2[/tex]
[tex]\\ \sf\longmapsto (1^3+2^3+3^3\dots k^3)+(k+1)^3[/tex]
Thus P(k+1) is true whenever P(k) is true.
Hence by the Principal of mathematical induction statement P(n) is true for [tex]\bf n\epsilon N.[/tex]
Note:-
We can solve without simplifying the Question .I did it for clear steps and understanding .
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Answer:
Step-by-step explanation:
[tex]1^3=1=\dfrac{1^2*(1+1)^2}{4} =\dfrac{4}{4} =1\\\\\\\displaystyle \sum_{i=1}^{n+1}\ i^3=(\sum_{i=1}^{n}\ i^3) + (n+1)^3\\\\\\=\dfrac{n^2*(n+1)^2}{4} +(n+1)^3\\\\\\=(n+1)^2*(\frac{n^2}{4} +n+1)\\\\\\=(n+1)^2*\dfrac{n^2+4n+4}{4} \\\\\\=\dfrac{(n+1)^2*(n+2)^2}{4}[/tex]
How do you add 1.7 plus 2 and 2/3
Answer:
4.37 (repeated)
Step-by-step explanation:
Answer:
[tex]\frac{131}{30}[/tex]
Step-by-step explanation:
1. Convert the decimal into a fraction by writing our number above a fraction line without the decimal sign.
2. Count the decimal places. Below the fraction lines write the multiple of 10 with that many zeros so that would be [tex]\frac{17}{10}[/tex]
3. Now write all numerators above the lcd (least common denominator) 30. So that would be [tex]\frac{51 + 60 + 20}{30}[/tex]
4. Calculate the sum of the positive numbers. The result is [tex]\frac{131}{30}[/tex]
5. Answer is [tex]\frac{131}{30}[/tex]
When solving the given equation, what is the first step?
(a)
Given: -3 + 2 = 4
А
Add 3 to both sides
B
Subtract 3 from both sides
с
Multiply -3 to both sides
D
Divide -3 from both sides
(b)
What is the solution to the give equation?
Given: -3 + x = 4
x = "write the answer (number only) in the answer box"
Answer:
(a) A
(b) x = 7
Step-by-step explanation:
(a) Given : -3 + 2x = 4
Step 1: Isolate the x to one side. To do so, we get rid of -3 by adding 3 to both sides. (-3 + 3 =0)
2x = 4 + 3
(b) -3 + x = 4
x = 4 + 3
= 7
Complete the slope-intercept form of this line.
y=-4x +
Answer:
It is y = -4x + 9
Step-by-step explanation:
General equation of a line:
[tex]y = mx + c[/tex]
For a y-intercept, c :
consider point (2, 1):
y is 1, x is 2 and m is -4
[tex]y = mx + c \\ 1 = (2 \times - 4) + c \\ 1 = - 8 + c \\ c = 1 + 8 \\ c = 9[/tex]
therefore, equation is:
[tex]y = - 4x + 9[/tex]
The complete form of the equation is y = - 4x + 9
The slope intercept form of the line is given as:
y = - 4x + c
Where c = intercept , we need to calculate, c
Take any given value of y and x and substitute into the equation, the solve for c
Take x = 1, y = 5
5 = - 4(1) + c
5 = - 4 + c
Add 4 to both sides
5 + 4 = - 4 + c + 4
9 = c
Hence, the complete form of the equation is y = - 4x + 9
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Answer the question with explanation;
Answer:
The statement in the question is wrong. The series actually diverges.
Step-by-step explanation:
We compute
[tex]\lim_{n\to\infty}\frac{n^2}{(n+1)^2}=\lim_{n\to\infty}\left(\frac{n^2}{n^2+2n+1}\cdot\frac{1/n^2}{1/n^2}\right)=\lim_{n\to\infty}\frac1{1+2/n+1/n^2}=\frac1{1+0+0}=1\ne0[/tex]
Therefore, by the series divergence test, the series [tex]\sum_{n=1}^\infty\frac{n^2}{(n+1)^2}[/tex] diverges.
EDIT: To VectorFundament120, if [tex](x_n)_{n\in\mathbb N}[/tex] is a sequence, both [tex]\lim x_n[/tex] and [tex]\lim_{n\to\infty}x_n[/tex] are common notation for its limit. The former is not wrong but I have switched to the latter if that helps.
Answer:
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]
General Formulas and Concepts:
Calculus
Limits
Special Limit Rule [Coefficient Power Method]: [tex]\displaystyle \lim_{x \to \pm \infty} \frac{ax^n}{bx^n} = \frac{a}{b}[/tex]Series Convergence Tests
nth Term Test: [tex]\displaystyle \sum^{\infty}_{n = 1} a_n \rightarrow \lim_{n \to \infty} a_n[/tex]Integral Test: [tex]\displaystyle \sum^{\infty}_{n = a} f(n) \rightarrow \int\limits^{\infty}_a {f(x)} \, dx[/tex]P-Series: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{1}{n^p}[/tex]Direct Comparison Test (DCT)Limit Comparison Test (LCT)Alternating Series Test (AST)Ratio Test: [tex]\displaystyle \sum^{\infty}_{n = 0} a_n \rightarrow \lim_{n \to \infty} \bigg| \frac{a_{n + 1}}{a_n} \bigg|[/tex]Step-by-step explanation:
*Note:
Always apply the nth Term Test as the first test to use for convergence.
Rules:
If [tex]\displaystyle \lim_{n \to \infty} S_n = 0[/tex], then the nth Term Test is inconclusive.If [tex]\displaystyle \lim_{n \to \infty} S_n = l[/tex] (some number l), then the series is divergent by the nth Term Test.Step 1: Define
Identify
[tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2}[/tex]
Step 2: Find Convergence
Substitute in variables [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} \rightarrow \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}[/tex]Expand: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2}= \lim_{n \to \infty} \frac{n^2}{n^2 + 2n + 1}[/tex]Evaluate limit [Special Limit Rule - Coefficient Power Method]: [tex]\displaystyle \lim_{n \to \infty} \frac{n^2}{(n + 1)^2} = 1[/tex]Compute [nth Term Test]: [tex]\displaystyle \sum^{\infty}_{n = 1} \frac{n^2}{(n + 1)^2} = \text{div}[/tex]∴ by the nth Term Test, the series diverges.
Topic: AP Calculus BC (Calculus I + II)
Unit: Convergence Tests
solve x+5y=11 and -x+4y=7 using elimination
Answer: (1, 2)
Concept:
There are three general ways to solve systems of equations:
EliminationSubstitutionGraphingSince the question has specific requirements, we are going to use elimination
Solve:
Given expressions
x + 5y = 11
-x + 4y = 7
Add both equations together to elimination [x] variable
(x + 5y) + (-x + 4y) = 11 + 7
x - x + 5y + 4y = 18
9y = 18
Divide 9 on both sides
9y / 9 = 18 / 9
[tex]\boxed{y=2}[/tex]
Find the x value
x + 5y = 11 ⇔ Given equation
x + 5(2) = 11 ⇔ Substitute values of y
x + 10 = 11 ⇔ Simplify by multiplication
x + 10 - 10 = 11 - 10 ⇔ Subtract 10 on both sides
[tex]\boxed{x=1}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Answer:
Stack the two equations and add them together as if they were multi-digit numbers. That will eliminate the x and then you can solve for the y. Knowing y you can then go back and solve for x.
Step-by-step explanation:
x + 5y = 11
-x + 4y = 7
---------------- "Add" the equations above to get the equation below.
9y = 18
Now you can divide both sides by 9 to see that y is equal to 2.
Now that you know that y = 2 you can substitute y in either of the two original equations. I'll use the first.
x + 5(2)= 11
x + 10 = 11
x = 1
100,000 less than five hundred sixty thousand,three hundred thirteen is __________.
Answer:
100,000>56,313 56, 56,313<100,000
PLEASE HELP NOW
write and solve an equation to find the value of X. please explain in details and show your work!
Answer:
Solution=96°+x+35°
=x+96°+35°
=x=96°-35°
=x=61°
Please help me I nned help
Answer:
length*breadth*height
Help Please!!!
Find lim h → 0 f(x+h)-f(x)/h for the given function and value of x.
f(x) = -4x -1, x = 5
Step-by-step explanation:
[tex]\displaystyle \lim_{h \to 0} \dfrac{[-4(x + h) - 1] - (-4x - 1)}{h}[/tex]
[tex]\displaystyle \:\:\:\:\:\:\:\:\:=\lim_{h \to 0}\dfrac{-4h}{h}[/tex]
[tex]\:\:\:\:\:\:\:\:\:=-4[/tex]
The point A(1,4) has been transformed using the composition T(-1,2) D2. Where is A at?
Answer:
(6, 6)
Step-by-step explanation:
T (-1, 2) = move down 1, right 2
(1, 4) turns into (3, 3)
Dilation scale factor = 2
Multiply x and y value by 2
(3, 3) = (6, 6)
State the gradient of the line 2y = 3 – 2x.
Answer:
Step-by-step explanation:
2y = 3-2x
y = -x + 3/2
Slope of line = coefficient of x = -1
One bolt of cloth will make shirts for 47 people. How many bolts are needed for 376 people?
Answer:
8
Step-by-step explanation:
the relationship of one "unit" (bolt) of cloth to 47 people makes this number (47 people) also an unit.
to answer the question we need to find out, how many units fit into the target.
how do we do that ? by division.
when we divide a/b, then we are actually asking, how many times does b fit into a.
so, how many bolts are needed for 376 people ?
well, how often does 47 fit into 376 ? how often can we build groups of 47 people out of the 376 ?
the answer
376 / 47 = 8
so, 8 bolts are needed.
Answer:
1 bolts = 47 people
so, 47+47+47+47+47+47+47+47= 376 people
The answer is 8 bolts
How far below the surface of the water is the top of a submerged mountain if the ocean floor depth is 12,500 feet and the mountain has a height of 10,190 feet?
Answer:
12500 - 10190 = 2310 ft.
Step-by-step explanation:
Find the value of x (give in steps)
Answer:
[tex]sum \: of \: exterior \: angle \: for \: a \: quadrilateral= 360 \\ x + 5x - 5 + 4x + 20 + 4x - 5 = 360 \\ x \: terms \: togather \\ x + 5x + 4x + 4x - 5 - 5 + 20 = 360 \\ 14x + 10 = 360 \\ 14x = 350 \\ x = 25 \\ thank \: you[/tex]
SOMEONE HELP ME WITH THIS QUESTION PLEASE!!!!
Answer:
E(-2, -1.5)Step-by-step explanation:
Coordinates of the points C and D:
C(1, 6), D(-3, -4)The point E has coordinates of (x, y).
The point E is at 3/4 of CD from C and 1/4 of CD from the point D.
Find coordinates of E:
x = -3 + 1/4(1 + 3) = -3 + 1 = -2y = -4 + 1/4(6 + 4) = -4 + 2.5 = -1.5Solve |3n +6| = 21
Answers:
n= -5 and n = 5
n= -9 and n = 9
n= -9 and n = 5
no solution
Answer:
a
Step-by-step explanation:
You have 2 cups of ketchup. You need 2 Tbspoons to make barbecue sauce. How many servings can you make?
Answer:
1 cup equals 16 tablespoons you have 2 cups so that is 32 tablespoons so you can make 16 servings. I think :)
hope this helps
Find the LCM of 17, 120 and 240.
If R = {(1,2) ,(2,3) ,(3,4) ,(4,5) } is subset of A x B , Find domain (A) and Range (B)
Answer:
hey
here is the attachment of your question
Answer:
[tex]thank \: you[/tex]
find the y-intercept and x-intercept of the line. 2x+3y= -14
Answer:
X-intercept: (-7, 0)
Y-intercept: (0, -14/4)
Step-by-step explanation:
To find the intercepts plugin 0 for one variable and solve
2x+3(0)=-14
x=-7
2(0)+3y=-14
y=-14/3
X^2 -24x-108=0 factorize
Answer:
(x-27)(x+4) = 0
Step-by-step explanation:
I think you have a typo, and meant
x^2 -23x-108
what are two factors of 108 that differ by 23?
x^2 -24x-108 does not factor with rational coefficients.
John ate 1/6 of a cake. Mike ate 1/12 of it. How much of the cake did the two boys eat?
Answer:
1/4
Step-by-step explanation:
1/6 + 1/12 = 1/4
Answer:
3/3
Step-by-step explanation:
1/6= to 2/12 and 1 /12 = 2 /6
Can someone plz solve the math problem i attached as a screenshot. Thx
Answer:
[tex]radius \: of \: first \: circle \: = \frac{3}{2} = 1.5cm \\ radius \: of \: \: second = \frac{4}{2} = 2cm \\ radius \: of \: third = \frac{6}{2} = 3cm \\ then \: ab \: length \\ = radius \: of \: first + diameter \: of \: second \: +r adius \: of \: third \\ = 1.5 + 4 + 3 \\ = 1.5 + 7 \\ = 8.5 cm\\ thank \: you[/tex]
the set of all natural numbers less than 15 write set build notation
Answer:
{x : x Σ N, x<15}
Step-by-step explanation:
Set of all natural number less than 14 = {x : x Σ N, x<15}
Which property of equality can be used to justify this step?
15 - 10x = 6x
+ 10x + 10x
————————
15 = 16x
A. Substitution Property of Equality
B. Summation Property of Equality
C. Addition Property of Equality
D. Subtraction Property of Equality
does this table represent a function
Answer:
yes
Step-by-step explanation:
Each input has one corresponding output.
What is the best description of the relationship in the scatterplot below?
Step-by-step explanation:
As x increases. f(x) increases so their is a positive linear association.
let me see mathematics genius help me answer this question.
Step-by-step explanation: Answer is
(i). 19% or 14/75
(ii) 28% or 7/25
We are asked to find multiple probabilities.
We got to find the number of combinations posible,
Use the combinations formula
[tex]c {}^{n} _r{?} = \frac{(r + n - 1) \: fractorial).}{r \: fractorial(n - 1)fractorial} [/tex]
For math this is read as,
if n choose r,( r+n-1)!/r!(n-1)!.
Where r is how many things we need from and n is the number of things we choose from.
We need 2 things and we have 24 objects to pick from.
So r=2 N equal=24
Which equal
25!/2!(23)!
Which equals
[tex]300[/tex]
So there are 300 possible combinations.
Using
For the 1st question, Since we are given two independent events, we can just multiply the number of good articles by major.
[tex]14 \times 4 = 56[/tex]
So this means the probability is
[tex] \frac{56}{300} = \frac{14}{75} [/tex]
Which is 19%
For the 2nd question, the can multiply the number of minor articles by major articles.
[tex]6 \times 4 = 24[/tex]
So the probability
is
[tex] \frac{24}{300} = \frac{8}{100} = \frac{2}{25} [/tex]
Which is equal to 8%
A cell phone company $500 for a new phone and $60 for a monthly plan. If C(t) is a rational function that represents the average monthly cost of owning the cell phone, what is the range of the function?
A. R: (0,500)
B. R: (60,560)
C. R: R
C: R: (-infinity, infinity)
HOW DO I SOLVE THIS?!
The range of a function is the set of possible values that can be obtained from the dependent variable. The range of the function is: [tex]R: (500, \infty)[/tex]
Given that:
[tex]Phone = \$500[/tex]
[tex]Monthly\ Plan = \$60[/tex]
Let the number of months be t. So, the function C(t) is calculated as follows:
[tex]C(t) = Phone + Monthly\ Plan \times t[/tex]
[tex]C(t) = 500 + 60 \times t[/tex]
[tex]C(t) = 500 + 60t[/tex]
The range is calculated as follows:
The smallest possible value of t is 0 i.e. when no monthly subscription is done.
So, we have:
[tex]C(0) = 500 + 60\times 0= 500 + 0 = 500[/tex]
And the highest is [tex]\infty[/tex] i.e. for a large value of t
So, we have:
[tex]C(\infty) = 500 + 60\times \infty= 500 + \infty = \infty[/tex]
Hence, the range of the function is:
[tex]R: (500, \infty)[/tex]
Read more about range of functions at:
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Sasha places an online order. In addition to the price of the item she orders, she pays 5% tax and a flat shipping fee of $15. The total cost of her order is $141. What is the price of the item she orders?
Answer:
$120Step-by-step explanation:
Let the price is x.
Set the equation and solve for x:
(x + 5%) + 15 = 1411.05x + 15 = 1411.05x = 126x = 126/1.05x = 120Answer:
$119.7
Step-by-step explanation:
Shipping is typically taken after tax so we can subtract shipping and find the tax.
141 - 15 = 126
126 * 0.05 = 6.3
Now we can subtract to find what she spent on items.
126 - 6.3 = 119.7
Best of Luck!