An arithmetic progression (A.P.) has first term 2 and an unknown positive common difference d. A geometric progression (G.P.) has first term 16 and an unknown common ratio r, where 0<r<1.
a. (i) The fourth term of the A.P. is equal to the second term of the G.P., while the seventh term of the A.P. is equal to the fourth term of the G.P. Write the two independent equations that arise from these conditions.
b. (ii) Solve the simultaneous equations obtained in part (i) to determine the values of d and r.
c. (iii) For the values of d and r found, investigate whether there exists any other positive integer k>7 for which the k-th term of the A.P. equals the k-th term of the G.P. Justify your answer clearly.