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Thanks for letting everyone know iloty.

The job you took on is indeed very time consuming.

Good luck with all the things you are doing and we see you when your life quietens down.

Hey guest! Make sure to include numbers! :)

Altitudes AD and BE of triangle ABC intersect at H. If angle BAC=54 and angle ABC=52, then what is angle AHB?

Angles AHB and EHD are congruent!

Angle AHB = 106º

Haha, thanks, GA, those are indeed some great potential usernames.  They're so great that even you haven't been able to decide which one for yourself, either.  :-)  Oh well, we'll just have to keep plugging away. 

_.

Let first term be a and diff be n. 8a+28n=244 and a+2n=20. Solving a=6,n=7, so 6 is our answer.

Oh goodness! Let x=1. We have (6+a)(14+b)=260. So 84+14a+6b+ab=260. I will leave the rest to you. Just plug in 2 for x and solve.

By the binomial theorem, \(\left(x-\frac{1}{x}\right)^5=x^5-\frac{1}{x^5}-5x^3+\frac{5}{x^3}+10x-\frac{10}{x}=x^5-\frac{1}{x^5}-5(x^3-\frac{1}{x^3})+10(x-1/x)\). Plugging, our expression is equivalent to \(x^5-\frac{1}{x^5}-5(x^3-\frac{1}{x^3})+20=32\rightarrow x^5-\frac{1}{x^5}-5(x^3-\frac{1}{x^3})=12\). Now we just need to find x^3+1/x^3. By the binomial theorem, \(x^3-\frac{1}{x^3}-3(x+1/x)=8\rightarrow x^3-\frac{1}{x^3}=14\). Plugging, \(x^5-\frac{1}{x^5}-70=12\rightarrow \boxed{x^5-\frac{1}{x^5}=82}\)

We use complementary counting: what we don't want. How many options for the first digit? 1,3,4,6,7,8,9=7. Second? 0,1,3,4,6,7,8,9=8. Third? 0,1,3,4,6,7,8,9=8. So 7*8*8=448 in total. There are 900 total 3 digit numbers, so 900-448=452. Ans.

WLOG let f(x) be linear. So f(x)=ax+b. So \(f(x+1)=a(x-1)+b=ax-a+b\rightarrow ax-a+b-ax-b=-a\). Note that this is constant, so f(x) must be quadratic. Let f(x)=ax^2+bx+c. So \(f(x+1)=a(x+1)^2+b(x+1)+c=ax^2+2ax+bx+a+b+c\rightarrow \)\(f(x+1)-f(x)=2ax+c\implies 2a=6\rightarrow \boxed{a=3} \)

tysm for the explanation 

prove:

n^3-n is divisible by 3    

                                                               n³ – n 

Factor out an n                                      (n) • (n² – 1)  

Factor n² – 1                                          (n) • (n + 1) • (n – 1)  

Rearrange terms                                     (n – 1) • (n) • (n + 1)  

Note that the expression consists of three consecutive integers.  

When you have three consecutive integers, one of them will be divisible by 3. 

If one of the multipliers is divisible by three, then the product is divisible by 3.  

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(A) There are 26 ways.

(B) There are 12 ways.

A = 4, B = -3, C = 5.

I do not think the first answer is correct   

   x must be = to or greater than 4 to avoid a negative square root .....

The domain is x >= 1.

9 =F + 4D,   
- 84 =F + 31D, solve for F, D

F =205/9 - This is the first term
D =- 31/9 - This is the common difference.

23rd term =205/9 + (- 31/9 * 22)
                 = - 53

y = 2 ( x^2-2.5x ) + 4         complete the square for x

y = 2 ( x-1.25)^2  + .875      now shift h,k as requested

y = 2 ( x-3.25) -.125  

Here is a graph:

https://www.desmos.com/calculator/dwmvmim5cg

Yesterday   10 t   = 15 f       or      t = 15/10 f

Today   15/10 f + 27 = 2 f

                f = 54   = 15 pound bags      t = 108 bags

                        total weight = 1890#

a=-11;b=0;c=(a+2)^3;b=b+c ;printb,a,c;if(b==- 1197, goto6, 0); a=a+2;if(a<10, goto2, 0)

OUTPUT: n = 6   and   x = - 9

First find the slope, m       m =  (y1-y2)/(x1-x2)    it does not matter which point you choose as 1 or 2   ( results are the same)

your equation will then be     y = mx + b     substitute EITHER one of the points (x,y) into the equation to compute b

then you have it    y = mx + b       (put in the m and the b that you found)

1/   Find FG using angle H and FH

2/   Find area of ΔEFH

3/   Find the area of the parallelogram BCEH

4/   Subtract area of ΔEFH from the area of BCEH

All this is more than a hint.