As posted yesterday:

(n+2) / n    *   ( n+3)/ (n+1)  *  (n+4) / (n+2) = 10      <==== Given 

Simplifies to 

1/n * (n+3) / (n+1) * (n+4) = 10 

then:

(n^2 + 7n + 12 )/ (n^2 +n) = 10        doing the fraction division results in:

9n^2 + 3n-12 = 0 

Use Quadratic Formula with    a = 9    b= 3    and c = -12     to find  n = 1      (or - 4/3  <=== does not work) 

So the first fraction would be 3/1 

First, find the slope as  (y1-y2) /( x1-x2) = (8-17)/(5 - -4) =  (-9)/(9) = - 1

Then the line in mx+b form is 

    y = (-1) x + b     sub in the values of either point to find b = 13

so the equation becomes 

    y = -x + 13     re-arrange 

  x +y = 13      divide thru by 13 

  1/13x  +  1/13 y  =1 

Hey, NotThatSmart has the right method, but accidentally used 100 instead of 120 as the second term :

(120+n)/(60+n)  =  (160+n)/(120+n)    cross multiply to get 

n^2 + 240n + 14400 = n^2 +220n + 9600 

20n = -4800

n = -240        would be the constant to add to the numbers ....

SO the geometric sequence would be 

-180        -120         -80      and the common ratio would be   -120/-180 =   2/3 

Thanx, Not That Smart !!   I check in once in a while !   ,,,,,

(n+2) / n    *   ( n+3)/ (n+1)  *  (n+4) / (n+2) = 10      <==== Given 

Simplifies to 

1/n * (n+3) / (n+1) * (n+4) = 10 

then:

(n^2 + 7n + 12 )/ (n^2 +n) = 10        doing the fraction division results in:

9n^2 + 3n-12 = 0 

Use Quadratic Formula with    a = 9    b= 3    and c = -12     to find  n = 1      (or - 4/3  <=== does not work) 

So the first fraction would be 3/1 

If John has 13 students he will not be able to form a rectangle using 'whole' students 

  Factors of 15 :    1   3  5  15

    he could make a 3 x 5 rectangle 

14  factors:      1 2 7 14     a 2 x 7 rectangle 

13  factors:     1 13        only a 1 x  13   rectangle (straight line ) 

See attached image: