CPhill

a1 + a3 + a5  =  7

a1 + (a1 + 2d) +( a1 + 4d)  =  7

3a1 + 6d  = 7

a2 + a4  = 6

(a1 + d)   + ( a1 + 3d )  =  6

2a1 + 4d  = 6

3a1 + 6d   =  7    mult through by 4

2a1 + 4d = 6      mult through by -6

12a1 + 24d  = 28

-12a1 - 24d  = -36         adding these we get

0  =  -  8

No solution

W               A   15      X

                                 m

                                 C

                                3m                            

Z         B      20         Y

tan  CAX =  m / 15   

tan  CBY  = 3m/ 20

Since angle ACB   = 60   then   angles CAX + CBY = 120

So

tan  ( CAX + CBY)  =   tan (120)

Identity  

tan (A + B)  =  [ tan A + tan B ] / [ 1 - tan A * tan B ] 

[ m/15 + (3m) / 20 ] / [ 1 - (m/15) [ (3m) / 20] ] =       -sqrt 3

Solving this  for positive m produces  m ≈  18.05  .....3m ≈  54.15

AC =  sqrt [ 15^2 + 18.05^2 ] ≈  23.47

BC =  sqrt [ 20^2 + 54.15^2 ] ≈ 57.73

Law of Cosines

AB  =  sqrt [ 23.47^2 + 57.73^2  - 2 ( 23.47 * 57.73)(1/2) ]  ≈  50.29

No

Because AD is parallel to BE.....then  because CE  is a transversal cutting  parallel lines, angle ADC  = angle BEC

But angle ACD  >  angle BCE

So  in the two triangles

ADC + ACD  + CAD  =  BEC  + BCE +  CBE

ACD  +  CAD  =  BCE  + CBE

Since  angle ACD  > angle BCE, then angle  CAD  must be  <  angle CBE

The only way that angle CAB = angle CBE  is  if A is on  BC

2T + 1B =  3.15     →  2T   = 3.15   - 1B    →   [3.15 - B ]  / 2  =  T         (1)

1M + 1B  = 3.50  →   M = 3.50 - B          (2)

1T + 2B + 3M  = 8.15     (3)

Sub (1) and (2)  into (3)

[ 3.15 - B ] / 2  +  2B  + 3 [ 3.50 - B ]  = 8.15               mutiply through by 2

[3.15 -  B ] + 4B  + 6 [ 3.50 -B ] = 16.30

-3B + 3.15 + 21  = 16.30

-3B  =  16.30 - 3.15 - 21

-3B = -7.85

B =  7.85 / 3  ≈  $2.62 = 262 cents

https://web2.0calc.com/questions/help-me_40934

6000 / 2000  =  x  /40

3 = x /40

120  = x  = 120 min

a2 + a4 + a6 + a8 + a10 + a12  = 1

a1 + a3  + a5 + a7 + a9 + a11  = -2          subtract these and we  get

Note :   a_n - a_{n -1}  =  d

d + d + d + d + d + d= 3

6d  =3

d = 1/2

Sum of terms  from  a1 to an  =  n * a1 +  (n)(n-1) / 2  *  d

S12  =  

12a1  +  66d =  3

12a1 +  66 (1/2)  =  3

12a1 + 33  = 3

12a1 =  -30

a1 = -30 / 12 =   - 5/2

Sum of an arithmetic series  from  a1 to an  inclusive =   n * a1  +  (n)(n-1) / 2 * d

S5 =      5a1 + 10d =   1/5         (1)

S10 =   10a1 + 45d =  1/10      (2)

Multiply  (1)  through by -2   and add it to (2)

25d = -3 /10

d =  -3 / 250

5a1 + 10 (-3/250)  = 1/5

5a1 + -30/250  = 1/5

5a1  = 1/5 + 3/25

5a1 = 8/25

a1 = 8/125

S15 =    15 ( a1)  +  105 d =    15(8/125) + 105 ( -3/250)   =  -3/10

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