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26 * r^3 = 12

r^3 = 26 /12  = 13 / 6

12 * r^6 =       32nd term

12 * (r^3)^2  =  

12 ( 13/6)^2  = 

12 ( 169  / 36 )  =

169 /  3  

We can solve this

[100 + x ]  / [ 60 + x] =  [ 165 + x ] / [100 + x]

[100 + x]^2 = [ 60 + x ] [ 165 + x ]

x^2 + 200x + 10000 =  x^2 + 225x + 9900

10000 - 9900  =  25x

100 =  25x

x =  100 /  25  =  4

Check

104 / 64 = 13 / 8

169 / 104  =  13 / 8

r  =13 / 8

Only the odd terms are positive

So

1000*(1/3)^(n -1 )  > 1      

(1/3)^(n-1)  >  1/1000  take the log of both sides

(n -1) log (1/3)  > log (1/1000)                   (log 1/3  is  negative....reverse the sign )

n -1 <  log (1/1000) /(log (1/3)

n < 1  + log (1/1000) /log (1/3)

n < 7.28

The 1st , 3rd, 5th and  7th terms will all be  > 1  ........4 terms

1/4  + 12d = 2/5

12d = 2/5 -1/4

12d =  3/20

d =  3/20 * 1/12  =  1/80

2/5 + 13d =  48th term

2/5 + 13 ( 1/80)  = 

[ 32]/80 + [13]/80  = 

45 / 80  = 

9 / 16

The numerator and  denominator are  defined  for all real  numbers.....so...

domain  =  ( -inf,  inf)

We need to solve this

C (n ,14)  >  100

By brute force

C (15,14)  =  15

C ( 16 ,14)    =  120

So......n =  16

1.  Since we have the condition "at least once," then "24" could  also appear twice, as well ...so... I think we could also have

2424x     =    10

24x24     =    10

x2424   =        9

= 19  more added to   blueorange's answer

Simplify

x^2 - mx + 14  = 0

Factors of 14                                                m

(1 , 14)                   (x + 1) (x + 14)              -14

(-1, -14)                  (x - 1)  (x - 14)               14

(2,7)                       (x + 2) ( x + 7)                -9

(-2, -7)                   (x - 2) ( x - 7)                   9

First one   C(18,2)

Second one :  Choose Mark and  1 of 17 other people =17 ways  

Lst one :  C(17,2)

A

      5

            X

               5

B                 C

AX =CX  . ..so ....BX  is also an angle  bisector .......so  AB = BC

Triangle AXB ≈  Triangle ABC

AX / AB  = AB / AC

5 / AB = AB / 10

AB^2  = 50

AB = 5sqrt (2) = BC

BX / AB  = BC / AC

BX / AB = AB  / AC

BX / AB = AB / 10

10BX = AB^2

10 BX = 50

BX = 50 / 10    =  5

For the first function x can  be any real number except 0

For the second function , we  have to  have that

5x + 6 - 11 ≥ 0

5x - 5 ≥  0

5x ≥  5

x ≥  1    [this is the domain ]

[ y  - - 2] / [ 6 -1] =  -7/4

[y + 2 ] = -(7/4) (5)

y + 2 = - 35/4

y =  -35/4 - 2

y = -43/4

Points (3,0)  and ( -1,-1) are on the  graph

Slope =  [ 0 - -1 ] / [ 3 - -1] = 1/4

Using (3,0)

y =(1/4) ( x -3)

y =(1/4)x - 3/4    multiply through by 4

4y = 1x -3

1x - 4y  = 3

Semi-perimeter =  [ 15 + 10 + 12 ]  / 2 =  18.5

Area  = sqrt  [18.5 * 3.5 * 6.5 * 8.5 ]  ≈  59.8

Area =  (1/2) (longest side )( altitude to this side)

 59.8  = (1/2) (15)  (altitude to this side)

59.8 / 7.5  ≈  7.97  =  shortest altitude

What is BX?

Angle B is 90°. Angle BXA is 90°.

So BX is the height of the Thales circle with radius 5.

\(\color{blue}\overline {BX}=5\)

 !

using 1/2 ab sin c formula we easily see that the answer is 12

Pick any two points, I chose (3, 0) and (-1, -1).

Calculate slope:

\(\frac{-1 - 0}{-1 - 3}=\frac{1}{4}\)

From slope-intercept form, you know the equation is

y = (1/4)x + b

Plug in any point to find b. I chose (3, 0)

0 = (1/4)*3 + b

b = -(3/4)

Therefore the equation is 

y = (1/4)x - (3/4)

To get into the desired form, subtract (1/4)x from both sides then multiply by lcm

-(1/4)x + y = -(3/4)

x - 4y = 3, is the equation in standard form.

Correct me if I'm wrong, but SSA congruence is not a thing, meaning the triangle is nonunique

Therefore there are multiple possible values for the area.

To find all possible combinations without ordering, we just set the number of 3s to be 1, 2, 3, or  4 and set the rest to 1s.

We can split into these following cases:

#1: 4 3s and 0 1s

#2: 3 3s and 3 1s

#3: 2 3s and 6 1s

#4: 1 3 and 9 1s

#5: 0 3s and 12 1s

Now we find the number of ways to order each case:

Case 1: 

1 way to order this:

Case 2:

\(\frac{6!}{3!3!}=20\) ways to order this

Case 3:

\(\frac{8!}{6!2!}=28\) ways to order this

Case 4:

\(\frac{10!}{1!9!}=10\) ways to order this

Case 5:

1 way to order this.

Adding, 1+20+28+10+1 = 60 total numbers.

We can find the slope of the line to find y.

First, we us points A and B to find the slope. 

Finding the slope, we have \(5-(-2)\over{-3-1 } \) which is equal to \(-{7\over4}\)

Then, applying the point slope formula, we get \(y-1=-{7\over4}(6+2)\).

Simplifying we get \(y=-13\)

Therefore, y = -13.

Find the length of the shortest altitude in this triangle.

\(a_a:a_b:a_c=\frac{1}{a}:\frac{1}{b}:\frac{1}{c}\\ a_a:a_b:a_c=\frac{1}{10}:\frac{1}{12}:\frac{1}{15}\\\)

\({\color{blue}The\ shortest\ altitude}\ in\ this\ triangle\ \color{blue}is\ a_c.\\ \overline{AC}=12\ ,\ \overline{BC}=10\ ,\ \overline{AB}=15\)

            C

             |

             \(a_c\)                      Triangle ABC

             |

A     x    .     15-x    B

\( a_c\ ^2=12^2-x^2=10^2-(15-x)^2\\ 144-x^2=100-(225-30x+x^2)\\ 144-100+225=30x\\ x=8.9\overline 6\)

\(a_c\ ^2=12^2-8.9\overline 6\ ^2\\ a_c=\sqrt{12^2-8.9\overline 6\ ^2}\\ \color{blue}a_c=7.97489\)

 !

Thanks