Answer
$$480 \cdot {5^{14}}{a^2}{b^{14}} + 32 \cdot {5^{15}}a{b^{15}} + {5^{16}}{b^{16}}$$
Work Step by Step
$$\eqalign{
& {\rm{last\, three \,terms\, of\, }}{\left( {2a + 5b} \right)^{16}} \cr
& {\rm{Apply\, the\, binomial\, theorem\, to\, the\, last\, four \,terms}} \cr
& {\left( {2a + 5b} \right)^{16}} = \left( \matrix{
16 \hfill \cr
14 \hfill \cr} \right){\left( {2a} \right)^2}{\left( {5b} \right)^{14}} + \left( \matrix{
16 \hfill \cr
15 \hfill \cr} \right)\left( {2a} \right){\left( {5b} \right)^{15}} + {\left( {5b} \right)^{16}} \cr
& {\left( {2a + 5b} \right)^{16}} = {{16!} \over {2!14!}}{\left( {2a} \right)^2}{\left( {5b} \right)^{14}} + {{16!} \over {1!15!}}\left( {2a} \right){\left( {5b} \right)^{15}} + {\left( {5b} \right)^{16}} \cr
& {\rm{Simplify}} \cr
& {\left( {2a + 5b} \right)^{16}} = 120\left( {4{a^2}} \right)\left( {{5^{14}}{b^{14}}} \right) + 32a\left( {{5^{15}}{b^{15}}} \right) + \left( {{5^{16}}{b^{16}}} \right) \cr
& {\left( {2a + 5b} \right)^{16}} = 480 \cdot {5^{14}}{a^2}{b^{14}} + 32 \cdot {5^{15}}a{b^{15}} + {5^{16}}{b^{16}} \cr} $$
