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
HELPOOPPPPPPPPPSBSBBSHSHABDBNSKA
NNADVOKAT [17]

Answer:

Step-by-step explanation:

5 0
3 years ago
PPlz help meeee need answer
user100 [1]
P^2 could also have -4 as an answer. Any time a number is squared, it is positive
5 0
3 years ago
What is the product of (x+2) (x-4)
skelet666 [1.2K]
(x + 2)(x - 4)

= x^2 + 2x - 4x - 8
= x^2 - 2x - 8
3 0
3 years ago
2x - 12 = 18 Algebra amazing huh​
Inessa [10]

Answer:

x=15

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
CAN SOMEONE HELP ME PLEASE ASAP!
mariarad [96]

Answer:

880 feet per hour.

Step-by-step explanation:

26 inches to foot = 2.2

9 seconds to hour =  0.0025

2.2/0.0025 = 880 feet per hour.

If you need any more help please let me know and i will make a video for you.

4 0
3 years ago
Other questions:
  • What is 363 round to the nearest ten and hundred
    7·1 answer
  • Sheyla goes shopping at the bookstore. She buys 2 DVDs for $8.25 each. Sheyla also buys two books at $5.00 each. How much change
    9·1 answer
  • Is zero less than equal to negative six
    14·2 answers
  • Solve 30 divided by 510
    13·2 answers
  • Please answer quickly, no explanation needed! (MathsWatch)
    13·2 answers
  • Add integers.<br> 0<br> -5+ (-3) =
    14·2 answers
  • Bradley paid $33.53 for 7 gallons of milk. What is the price per milk gallon
    6·2 answers
  • I would really appreciate it if someone could help and answer this with an explanation:)
    11·1 answer
  • FrEEEEEEEEEEEEEEEEEEEEEEEEEEE Points!!!!!!!!!!!!!!!!!
    8·2 answers
  • Which list of angle measures could be the angle measures of a triangle?
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!