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4 years ago

15

Am I correct?? Will give brainliest!!!

1 answer:

Rashid [163]4 years ago

3 0

Yes, However the first one is:

1. Distributive Property

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

Molodets [167]

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.

7 0

3 years ago

I don't understand this plz help

Ivanshal [37]

Hey there! :D

When you are finding the area of corresponding sides, you square the number of the ratio of the similar lengths.

The screens a little blurry, but I'm assuming the corresponding side is represented as:

$\frac{a}{b}$

So, you would keep a and b where they are, (you would be confusing the areas if you switched it in any way) and then just square the variables.

$\frac{a^2}{b^2}$

"C" is the best answer.

I hope this helps!
~kaikers

3 0

3 years ago

Why is the circumference formula more accurate than actually measuring around the circle?

goblinko [34]

Answer:

Because a circle has infinite sides and it is almost impossible to be very accurate measuring around.

5 0

3 years ago

Use the associative property to find more factors of 12 and 30

ahrayia [7]

The associative property could be that they are both even and they are both in the 3x table

8 0

3 years ago

How do I do it and be able to understand it

poizon [28]

Do what ? post for i can help you

3 0

2 years ago

Read 2 more answers