CPhill

https://web2.0calc.com/questions/help-triangles_2

 {   an  =  n* a1  +  (n) (n-1) / 2 * d   }

23a1 + 253d = 1/4   mult through by  33   →   759x + 8349y = 33/4 

33a1 + 528d = - 1/5 mult through by  -23  →    759x - 12144y = 23/5

Add these

-3795d  = 257 / 20

d = -257 / 75900

23a1 +  253 (-257/75900)  = 1/4

a1 = 83  /1725

a43  = 43(83/1725)  +  903(-257/75900)  =   -15007 / 15180

1000 (-8/9)^(n -1)  > 1

(-8/9)^(n -1)   is  only positive when  n  is odd

So we can  write

1000 (8/9)^(n -1)  >  1

(8/9)^(n - 1) > 1/1000     take the log of both sides  and by a log property we can write

(n -1) log (8/9) > log (1/1000)        

{ divide both sides by log (8/9)....this is negative so reverse the inequality sign }

n - 1  <   log (1/1000) / (log (8/9)

n <  1 + log (1/1000) /log (8/9)

n < 59.6

So 

n = 59

The number of terms > 1 =   [ 59 + 1 ] / 2  =  30

3x + 2y + xy = 115 + 11x + 2y    simplify

8x - xy = - 115

x ( 8 - y)  =  - 115

x ( y - 8)  =  115

factors of 115  =   1   5   23    115

Four possible solutions  for x, y

y = 31    x =  5

y = 123  x = 1

y = 13    x = 23

y = 9     x = 115

x + y =   36  or  124

a1 + a3 + a5 =  22

a1 + (a1 + d) + (a1 + 2d)   =22

3a1 + 3d =  22   

a2 + a4 =  17

(a1 + d) + (a1 + 3d)  = 17

2a1 + 4d = 17

3a1 + 3d  = 22   mult through by 4  →   12a1 + 12d  =  88

2a1 + 4d =  17   mult through by -3 →     -6a1 - 12d  = -51       add these

6a1  =  37

a1  = 37 / 6

x + y  = 2

square both sides

x^2 + 2xy + y^2  = 4

x^2 + y^2 =  4 - 2xy

xy = 5  -x^2 - y^2

xy = 5 - (x^2 + y^2)

So

x^2 + y^2  = 4 -2xy

x^2 + y^2  = 4 - 2 [ 5 - (x^2 + y^2) ] 

x^2 + y^2  =  4 - 10 + 2(x^2 + y^2)

-3(x^2 + y^2)  =  -6

x^2 + y^2  = -6 / -3  = 2

a + b = 4

square both sides

a^2 + 2ab + b^2  =16  →  a^2 + b^2  = 16 - 2ab

a^2 + b^2 = 6

So

6 = 16 - 2ab

-10 = -2ab

5 = ab

-5 = -ab

a^3 + b^3  =  

(a + b) ( a^2 + b^2 - ab) =

(4) (6 -  - 5)

(4) (11)  = 

44

Simplify as

8a + b = -3

9a + (2 +m)b = k 

This will have infinte solutions when the equations are the  same...so....multiply the first equation  by 9 and the  second by 8 and  we  get

72a + 9b = -27

72a + 8 (2 + m)b = 8k

So

9 = 8(2 + m)            and          8 k  = -27

9 = 16 + 8m                              k = -27/8                           

-7 = 8m

m = -7/8

1.  ax^2 + bx + a

Product of the roots =  a/a   = 1

So....call the second root, m

And

m * 4  = 1

m =  1/4 = the other root

Let 2.57....  = x

Let  257.57... = 100x

100x   =   257.57....

-    x       -   2.57.....   

__________________________

99x  =     255

x = 255 / 99 =  85 / 33