The number of terms common between the series 1 + 2 + 4 + 8 + . . . . . to 100 terms and 1 + 4 + 7 + 10 + . . . . . to 100 terms is

Solution :
For G.P., $${t_n} = {2^{n - 1}};$$   for A.P., $${T_m} = 1 + \left( {m - 1} \right)3.$$    They are common if $${2^{n - 1}} = 3m - 2\,\,{\text{or, }}{{\text{2}}^{n - 2}} + 1 = \frac{{3m}}{2} \leqslant 150$$
$$ \Rightarrow \,\,n \leqslant 9,m \leqslant 100.$$
By trial, $$n = 1, m = 1; n = 3, m = 2; n = 5, m = 6; n = 7, m = 22; n = 9, m = 86.$$