SOLUTION: find k so that 2k+1, 3k+1 and 5k+1 form a geometric sequence

For it, you should have


     = ,   or

     = ,   or, cross multiplying

    (5k+1)*(2k+1) = ,   or, simplifying

    10k^2 + 7k + 1 = 9k^2 + 6k + 1

    k^2 + k = 0

    k(k+1) = 0

    k= 0  OR  k= -1.


At k= 0,  the three terms of the GP are 1, 1, 1.


At k = -1,  the three terms of the GP are -1, -2, -4.


So, there are 2 possibilities.


One geometric progression is  {1, 1, 1,}.  It corresponds to the value of k= 0.


Another geometric progression is  {-1, -2, -4}.   It corresponds to the value of k= -1.