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nice!

a1 =  (2(1))+1 = 3

a10 = 2(10)+ 1 = 21

sum =  n/2 ( a1 + a10) = 10/2(3 +21) = 5 * 24 = 120

\(\displaystyle\sum_{n=1}^{10} \mbox{ } (2n + 1) = (2 \cdot 1 + 1) + (2 \cdot 2 + 1) + \mbox{ }... \mbox{ } + (2 \cdot 10 + 1) = 120\),

for example, when n = 2 (the number under the summary sign), then start with 2 till the number over the summary sign. If you don't get it, e.g.

\(\displaystyle\sum_{n=2}^{5} \mbox{ } n + 1\) you see n = 2, so (2 + 1), then comes n = 3, so (2 + 1) + (3 + 1) + ... (5 + 1) = 18,

so \(\displaystyle\sum_{n=2}^{5} \mbox{ } n + 1 = 18\) .

Feel free to ask if you still don't get it.

Gino is right,   

You have copied the question incorrectly.

ABCD  is a regular tetrahedron (right pyramid whose faces are all equilateral triangles).

If M  is the midpoint of line segment CD , then what is cos

See if you can make sense of my pic and then use cosine rule.

[1991*1993*1995*1997*1999*2001*2003]  mod 10^10==3846571165

1 + 6 + 5 ==12

Find the inradius of triangle ABC if it's sides lengths are 29, 29, 40.

Hello Guest!

a = b = 29     c = 40

\(cos(\alpha)=\frac{c}{2\cdot b}=\frac{40}{2\cdot 29 }\\ \alpha =46.397^o\\ f(x)=tan( \frac{\alpha}{2})x\\ x=\frac{c}{2}=20\\ r=tan( \frac{\alpha}{2})\cdot x=tan(\frac{46.397^o}{2})\cdot 20 \)

\(r=8.\overline{571428}=\large \frac{60}{7}\)

  !

\(\frac{1}{x} + \frac{1}{y} = \frac{1}{18} + \frac{1}{3} = \frac{7}{18}\)

\(\frac{x+y}{xy} = \frac{7}{18}\)

\(18x + 18y = 7xy\)

\(0 = 7xy - 18x - 18y\)

\(324/7 = (7x - 18)(y-18/7)\)

\(324 = (7x - 18)(7y - 18)\)

Create a table with the factors of 324, which is what (7x - 18) and (7y - 18) will multiply to. By brute force, you can find the smallest value of x + y.

simplify the quadratic:

\(2x^2 - 3x + 27 + 3x^2 + 5 = 5x^2 - 3x + 32\)

Let the roots be x and y. From Vieta's formulas, we know that:

sum of roots = x + y = 3/5

product of roots = xy = 32/5

sum of squares of roots = x^2 + y^2 = (x + y)^2 - 2xy = (3/5)^2 - 2(32/5) = -12.44

\(f(x) + bg(x)\)

\(= (x^4 - 3x^2 + 2) + b(2x^8 - 2x^4 - 6x^2 + 2x - 1)\)

\(= 2bx^8 + (-2b + 1)x^4 + (-6b - 3)x^2 + 2xb + (2-b)\)

If b = 0, then the x^8 term is eliminated, and the smallest possible degree of the polynomial is 4.

g(x) = f(f(x)) - f(f(f(x)))

= f(3x+2) - f(f(3x+2))

= 3(3x+2) + 2 - f(3(3x+2) + 2)

= 9x + 8 - f(9x + 8)

 = 9x + 8 - (3(9x + 8) + 2)

= 9x + 8 - (27x + 26)

= -18x - 18

x is in [0, 2]

g(x) is in [-54, -18]

20 consonants, 6 vowels,

3 letter words, at least one vowel

All letters different     6*25*24*3! = 900

2 vowels the same    6*25*3       = 450

2 consonants the same 6*20*3  =  360

all three the same       6             =  6

900+450+360+6 = 1716

Feel free to aks questions.

If you know combinatorics, you will see that this is simply 11 choose 3. 

For those of you who don't, the formula for choosing a number out of another number is n! / ((n – r)! r!), where n = number of items and c = how many you choose.

11 choose 3 = 11! / (11-3)! * 3! = 11! / 8! 3! = 11*10*9 / 3*2*1 = 11*5*3 = 165 ways.

(Alphabetical order does not matter here because you could always reorder them to be alphabetical and combinatorics do not count each combination more than one time)

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ShockFish

The two equations are: \(y = 2x+1\) and \(y - 3 = 2x - 2\). The first one is already in slope-intercept form, but the second one is not. We can simplify the second one to \(y= 2x+1\). Because the \(2\) equations are identical, they intersect at an \(\infty\) amount of points. This means that \(\color{brown}\boxed {t = \infty}\)

We know that 21% = 21/100, and "of" means multiplication, so:

21/100 * 121 = 21 * 1.21 = 25.14.

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ShockFish

First, let's simplify it: 

|5a + 2| < 9

For an absolute value inequality, you split it into two - one is positive and the other is negative. (Remember to flip the sign in the negative one!)

5a + 2 < 9

&

5a + 2 > -9

For the first one: 

5a < 7

a < 7/5

For the second one: 

5a > -11

a > -11/5

Thus we have our range: -11/5 < a < 7/5

Therefore, in interval notation, our answer is (-11/5, 7/5).

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ShockFish

Let's call the unknown number \(x\). By subsituting, we have \((7t-10)(3t+x)\). If we simplify this equation, we get \(21t^2+7tx-30t-10x\). 

A binomial must have exactly \(2\) terms, so \(7tx-30t\) will have to equal \(0\). 

The only way this can be possible is if \(\color{brown} \boxed {x=4 {2\over 7}}\)

The sum of the angles in a convex polygon is \(180\times(s-2)\). If we use \(4560\) as the sum of all the angles in the polygon, we have the equation \(180\times(s-2)=4560\). This simplifies to \(s=82/3\). Now, we have to round up, meaning that there is \(\color{red}\boxed {28}\) sides in the polygon.

What's the figure?

Sum of the 52 entries in the series  is     52/2 ( 2 + 104)  =  2756      2756 mod 7 = 5

We are given two points: (x, y) = (12, 2.5) and (16, 3.5)

We can find the slope of this line:

\(\frac{3.5 - 2.5}{16 - 12} = \frac{1}{4} = 0.25\)

Now we can find the y-intercept. First, pick a random point:

y = mx + b

2.5 = (0.25)(12) + b

b = -0.5

The linear function is y = 0.25x - 0.5.

To find the missing y-value, plug in x = 8 to get y = 0.25(8) - 0.5 = 1.5

there are two options: 1) he did the calculations and got it wrong or 2) he guessed and got it wrong.

1) 1/4 * 1/5 = 1/20

2) 3/4 * 2/3 = 1/2

The probability we want is the answer for #1 divided by the sum of #1 and #2.

\(\frac{1/20}{1/20 + 1/2} = \frac{0.05}{0.55} = \boxed{\frac{1}{11}}\)

ratio of 3:2 -> Adriana's income is 3x and Belen's income is 2x

3x + 2x = 90000

5x = 90000

x = 18000

Adriana should receive 3x = $54000.

The inradius can be found using the formula A = rs, where A is the triangle's area, r is the inradius, and s is the triangle's semiperimeter.

\(s= \frac{29+29+40}{2} = 49\)

\(A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{(49)(20)(20)(9)} = 420\)

\(\rightarrow r = \frac{A}{s} = \frac{420}{49} = \boxed{\frac{60}{7}}\)

We can make a Venn Diagram:

[baseball || both || football] neither

[     37     ||   x    ||      26   ]    24

37 + 26 - x + 24 = 75

x = 12

1. either not on the football team OR on the baseball team

The set of people on the baseball team is a subset of the people not on the football team.

probability = (37 + 24)/75 = 61/75

2. at least on the baseball team OR not on the football team

This problem has the same answer as #1. (The wording is kind of confusing; I don't think "at least... or" has any difference from "either... or".)

3. either on the football team OR not on the baseball team

The set of people on the football team is a subset of the people not on the baseball team.

probability = (26 + 24)/75 = 50/75 = 2/3

4. at least on the football team OR not on the baseball team

This problem has the same answer as #3.

5. neither on the football nor the baseball team

This information was given in the problem.

probability = 24/75 = 8/25