\[a_{1} + a_{2} = 25;\ \ a_{2} + a_{3} = 39.\]
\[1)\ a_{1} + a_{1} + d = 25\]
\[2a_{1} + d = 25\]
\[2a_{1} = 25 - d.\]
\[2)\ a_{1} + d + a_{1} + 2d = 39\]
\[2a_{1} + 3d = 39\]
\[2a_{1} = 39 - 3d.\]
\[3)\ 25 - d = 39 - 3d\]
\[- d + 3d = 39 - 25\]
\[2d = 14\]
\[d = 7.\]
\[4)\ 2a_{1} = 25 - d = 25 - 7\]
\[2a_{1} = 18\]
\[a_{1} = 9.\]
\[a_{2} = 9 + 7 = 16;\]
\[a_{3} = 16 + 7 = 23.\]
\[Наибольшее\ число:23.\]
\[Ответ:23.\]