here's the solution

 

As per the qn, there are 6t terms in an A.P. S1 : sum of first 2t terms

                                                                S2 : sum of first 4t terms

                                                                S3 : sum of last 2t terms

 

 given 3n terms of an A.P.,

Sum of 1st n terms, Sum of next n terms, Sum of last n terms are in A.P.

 

Then here S1, (S2 - S1), S3 are in A.P.  ---------- (1)

 

Also given, (S1*S2*S3 - S1^2*S3) = 4;

ie, S1*(S2 - S1)*S3 = 4

 

A.M. >= G.M.

 

{ S1 + (S2 - S1) + S3 } / 3  >=  cuberoot(S1*(S2 - S1)*S3)

                                        >=  cuberoot(4)

From eqn(1)  3(S2 - S1) / 3 >=  cuberoot(2^2)

                      (S2 - S1)     >=  2^2/3   

SO : (S2 - S1) not less than 2^2/3 .........