CPhill

Call the first term, N

Let the common ratio  be , r

So

Nr^3   =  2017

N    =      2017  / r^3

So  we  have  that

( 2017/r^3) ( 2017/r^2) ( 2017/r) (2017) (2107 r)  =  1024   simplify

(2017)^5  / r^5   = 1024

(2017)^5  /1024  =   r^5               take the 5th root of both sides

2017/4   = r

The second term  is

First term * r  = 

 [ 2017 / ( (2017/4)^3 ]  *  (2017/4)    =    16/2017

Call the common ratio , r

(x  - 2)r  = ( x + 2)     ⇒   xr   - 2r  =   x +  2             (1)

( x +2)r  = x +  4      ⇒    xr  + 2r  =   x +  4              (2)

Subtract  (1)  from (2)

4r  =   2

r  = 2/4 =  1/2

Using the  first equation

x (1/2)  - 2(1/2)   = x +  2

(1/2)x - 1   = x +  2

-(1/2)x  =  3

x  =  3/(-1/2)    = -6

(x - 2)  =     -8

( x + 2)  =   -4

(x + 4)   =   -2

Three friends Adam, Betty, and Christian took turn driving cross country from Sacramento, California to Philadelphia, Pennsylvannia. Each drove for the exact amount of time, but at different average speed due to the varying landscapes. Adam drove at an average speed of 55 miles per hour through the tricky Sierra mountain range. Betty took the second leg through the relatively straight-forward (and boring) part of the trip, at an average speed of 85 miles per hour. Christian ran into some urban traffic and only managed a speed of 70 miles per hour. Together, they covered 2800 miles. What was their total driving time of the whole trip, in hours?

Let the time each drove =  T

So    rate  *  time  = distance.....so....

55T  +  85T  +  70 T   =   2800

210T  =  2800

T  =2800/210    =   40/3  hrs  each

So.....the whole trip  took   3 (40/3)   = 40 hrs

In triangle UYV , angle  YUV  = (1/2) 80  =  40°

And anlge  YVU =  (1/2) 35  =17.5°

So angle UYV   = 180   - 40  -17.5    = 122.5°

And angle XYZ  is a  vertical angle to  UYV so it also measures 122.5°

Let  the last term be    1 + 4(n  -1)     where  n is the  total number  of terms

Sum of an arithmetic series = 

(first term + last term )  (number of terms   /2)

So we have

( 1   +1 + 4(n - 1) )  (n/2)   = 120           simplify

( 4n  -2)  (n/2)  =120   

(2 n -1)  n   =120

2n^2  - n  =120

2n^2  - n    -120    =  0   

(2n  + 15)  ( n  - 8)    = 0

The second factor  set  to   0  and solved for n gives us  our  answer  = 8 terms

sqrt (45)  = 3sqrt (5)

sqrt  (360) / sqrt (2)  =  sqrt(360/2)  = sqrt (180) =  sqrt (36 * 5)  =  6sqrt (5)

Combining  all of these    =      3sqrt (5)  +2sqrt (5)   + 6sqrt (5)  =  11sqrt(5) =  sqrt ( 121 * 5)  =

sqrt (605) 

N  =  605

The percents  of the  mixture  don't matter.......the same mixture percents will  be  used  no matter  what  their cost

If P  was  the price  before the increases

After the increases  the price  is   P (1.2)(1.25)   = 1.5P

So   the  price  has  increased   by 50%

Let the time Jennifer walks =  J

4 hrs * 12mph   = 48 miles

The total distance  she travels =  48 + 2J

So  the total hrs  she travels =  4 + J

The time it took   Thomas to reach the bike  is     48/ 4   = 12 hours

Let his time on the  bike =  T

So  the   distance  he travels  is    48 +  12T

And the  time he travels =  12 + T

The time they  travel is equal    and  so is the  distance  ....so...

4 + J  =  12  + T          (1)

48 + 12T  =  48 + 2J      ⇒     12 T  = 2J    ⇒   6T   = J      (2)

Sub (2)  into (1)

4 + 6T =   12 + T

5T  = 8

T  =  8/5   =   1. 6 hrs

The total time to catch  up to Jennofer  =      12 + 1.6  =  13.6  hrs

Check

Distance  Jennifer travels =   48 + J  =   48 + 2*6(1.6)  = 67.2 miles

Distance  Thomas  travels  =  48 + 12 (1.6)  =   67.2 miles

OOPS....small error....now corrected  !!!

The sum  = 48 ....   so.....

P + Q  =   44

P + R  =   38

P + S =    32

Q + S  =   26

Q + R   =  32

R + S   =  20

Subtract the first two  equations ⇒

Q - R  =  6

Q + R  =  32       add these

2Q =  38

Q =  19

S  =  7

P  =  25

R  =   13

P +   Q  +  R  +  S   =      25  +  19 +  13   +  7    =      64