What measure is equivalent to 65 killograms?

I need help with 2,3 and 6! Please Help!

Georgia [21]

2. If you already know Faulhaber's formula, which says

$\displaystyle\sum_{k=1}^nk^2=\dfrac{n(n+1)(2n+1)}6$

then it's just a matter of setting $n=4$ . If you don't, then you can prove that it works (via induction), or compute the sum by some other means. Presumably you're not expected to use brute force and just add the squares of 1 through 4.

Just to demonstrate one possible method of verifying the formula, suppose we start from the binomial expansion of $(k-1)^3$ , do some manipulation, then sum over $1\le k\le n$ :

$(k-1)^3=k^3-3k^2+3k-1$
$\implies k^3-(k-1)^3=3k^2-3k+1$
$\implies\displaystyle\sum_{k=1}^n(k^3-(k-1)^3)=\sum_{k=1}^n(3k^2-3k+1)$

The left side is a telescoping series - several terms in consecutive terms of the series will cancel - and reduces to $n^3$ . For example,

$\displaystyle\sum_{k=3}^2(k^3-(k-1)^3)=(1^3-0^3)+(2^3-1^3)+(3^3-2^3)=3^3$

Distributing the sum on the right side across each term and pull out constant factors to get

$\displaystyle n^3=3\sum_{k=1}^nk^2-3\sum_{k=1}^nk+\sum_{k=1}^n1$

If you don't know the formula for $\displaystyle\sum_{k=1}^nk=\frac{n(n+1)}2$ , you can use a similar trick with the binomial expansion $(k-1)^2$ , or a simpler trick due to Gauss, or other methods. I'll assume you know it to save space for the other parts of your question. We then have

$\displaystyle n^3=3\sum_{k=1}^nk^2-\frac{3n(n+1)}2+n$
$\implies\displaystyle\sum_{k=1}^nk^2=\dfrac{n(n+1)(2n+1)}6$

and when $n=4$ we get 30.

3. Each term in the sum is a cube, but the sign changes. Recall that $(-1)^n$ is either 1 if $n$ is even or -1 if $n$ is odd. So we can write

$1^3-2^3+3^3-4^3+5^3=\displaystyle\sum_{k=1}^5(-1)^{k-1}k^3$

( $k+1$ as the exponent to -1 also works)

6. If $0\le k\le n$ , and $i=k+1$ , then we would get $1\le k+1\le n+1\iff1\le i\le n+1$ . So the sum with respect to $i$ is

$\displaystyle\sum_{k=0}^n\frac{k^2}{k+n}=\sum_{i=1}^{n+1}\frac{(i-1)^2}{i+n-1}$

4 0

3 years ago

your class of 32 students is divided into fourths to create groups to answer a set of questions together then 1/4 of each group

Alecsey [184]

Answer:

2 students

Step-by-step explanation:

32 students are divided into fourths:

32/4 = 8

Each group has 8 people.

1/4 of the students in each group is the same as dividing 8 into fourths:

8/4 = 2

2 students from each group will go up to the front of the classroom.

Hope this helps :)

3 0

3 years ago

Please help!!!!!!!!!!!!!!

Montano1993 [528]

<h3>Answer: 7.1</h3>

=========================================================

Explanation:

The two points are (x1,y1) = (1,3) and (x2,y2) = (8,4)

Use the distance formula to get,

$d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}\\\\d = \sqrt{(1-8)^2 + (3-4)^2}\\\\d = \sqrt{(-7)^2 + (-1)^2}\\\\d = \sqrt{49 + 1}\\\\d = \sqrt{50}\\\\d = \sqrt{25*2}\\\\d = \sqrt{25}*\sqrt{2}\\\\d = 5\sqrt{2}\\\\d \approx 7.0710678\\\\d \approx 7.1$

Alternatively, you can plot the two points A(1,3) and B(8,4) on the same xy grid. Then plot point C(8,3). Note that C has the same x coordinate as B, and the same y coordinate as A. Triangle ABC is a right triangle, where the 90 degree angle is at point C. The distance from A to B is the same as finding the length of AB, which is the hypotenuse.

This means we can apply the pythagorean theorem to find the hypotenuse AB. The distance formula is a modified version of the pythagorean theorem. Refer to the diagram below.