1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
baherus [9]
3 years ago
8

Evaluate 1^3 + 2^3 +3^3 +.......+ n^3

Mathematics
1 answer:
Molodets [167]3 years ago
7 0

Notice that

(n+1)^4-n^4=4n^3+6n^2+4n+1

so that

\displaystyle\sum_{i=1}^n((n+1)^4-n^4)=\sum_{i=1}^n(4i^3+6i^2+4i+1)

We have

\displaystyle\sum_{i=1}^n((i+1)^4-i^4)=(2^4-1^4)+(3^4-2^4)+(4^4-3^4)+\cdots+((n+1)^4-n^4)

\implies\displaystyle\sum_{i=1}^n((i+1)^4-i^4)=(n+1)^4-1

so that

\displaystyle(n+1)^4-1=\sum_{i=1}^n(4i^3+6i^2+4i+1)

You might already know that

\displaystyle\sum_{i=1}^n1=n

\displaystyle\sum_{i=1}^ni=\frac{n(n+1)}2

\displaystyle\sum_{i=1}^ni^2=\frac{n(n+1)(2n+1)}6

so from these formulas we get

\displaystyle(n+1)^4-1=4\sum_{i=1}^ni^3+n(n+1)(2n+1)+2n(n+1)+n

\implies\displaystyle\sum_{i=1}^ni^3=\frac{(n+1)^4-1-n(n+1)(2n+1)-2n(n+1)-n}4

\implies\boxed{\displaystyle\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}4}

If you don't know the formulas mentioned above:

  • The first one should be obvious; if you add n copies of 1 together, you end up with n.
  • The second one is easily derived: If S=1+2+3+\cdots+n, then S=n+(n-1)+(n-2)+\cdots+1, so that 2S=n(n+1) or S=\dfrac{n(n+1)}2.
  • The third can be derived using a similar strategy to the one used here. Consider the expression (n+1)^3-n^3=3n^2+3n+1, and so on.
You might be interested in
3x + 5y = 7<br> -3x - 2y = -1
Oksanka [162]

Answer:

(-1,2) y=2 x=-1

Step-by-step explanation:

This is a system of equations and in this you see there is 3x and negative three x. To solve this you need tp get rid of one of the variables. so we can add the two equations together to get 3y=6. This means that y=2 if we divide by 3 on both sides. Then we just plug 3 in for y and we get 3x+10=7. We subtract 10 on both sides to get 2x=-3 which means x=-1.

6 0
2 years ago
A book with 650 pages has 3 pages with diagrams.
Mrac [35]

The pages that have diagrams is an illustration of proportions

0.46% pages have diagrams

<h3>How to determine the percentage</h3>

The given parameters are:

Total = 650 pages

Diagrams = 3 pages

The percentage of page that has diagram is then calculated as:

Diagram\% = \frac{Diagrams}{Total} * 100\%

So, we have:

Diagram\% = \frac{3}{650} * 100\%

Evaluate the product

Diagram\% = \frac{300}{650} \%

Divide

Diagram\% = 0.46 \%

Hence, 0.46% pages have diagrams

Read more about proportions at:

brainly.com/question/1781657

3 0
1 year ago
The school track team earned $2 for each cap they sold and $5 for each sweatshirt they sold. The total profits were $317. If the
Feliz [49]
The answer is 45 bc if they sell 46 that gives u the answer
4 0
2 years ago
Expanded form for 48,243. (
Dima020 [189]
(4*10,000)+(8*1,000)+(2*100)+(4*10)+(3*1)

40,000+8,000+200+40+3=48,243

That is one way. You can also do this using exponential form!

Hope that helps

4 0
3 years ago
Examine the following graph, where P(x) is the preimage and I(x) is the image after a transformation.
Arisa [49]

Answer:

the 4th answer I(x) = P(x-4)

Step-by-step explanation:

the first 2 options are out, because just adding values to the functional result values changes the y value, and the curve is pushed up or down. but not to the side.

just imagine

I(x) = P(x-4) means that the function value of previously at x-4 happens now at x. 4 units "later" (or more to the right).

"x+4" would make the functional value at previously x+4 happen now at x. so, everything would shift to the left (happen "earlier").

therefore, the 4th answer option is correct.

3 0
1 year ago
Other questions:
  • A baker uses a coffee mug with a diameter of 8cm to cut out circular cookies from a big sheet of cookie dough. What is the area
    11·1 answer
  • PLEASE HELP ASAP Which postulate or theorem proves that these two triangles are congruent
    13·2 answers
  • What is 1/100 of 605
    15·2 answers
  • Please help. look at the picture.​
    11·1 answer
  • Which polynomial is in standard form? 2xy^3 +3x^3 y^4 -4x^4 y^5 +9x^5 y^6 Which polynomial is in standard form?
    13·1 answer
  • HELP I NEED ANSWERS NOW!!
    9·1 answer
  • jessica has 12 red beads,18 yellow beads and 42 blue beads. She puts all the beads into a few jars. The number of beads of every
    13·1 answer
  • Find the distance between A and B.
    6·2 answers
  • If mLP = 17 in., and mNP = 4 cm, mLN = ____ ??
    9·1 answer
  • Log(2t+4)=log(14-3t)
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!