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The Adams–Moulton methods with ''s'' = 0, 1, 2, 3, 4 are ({{harvnb|Hairer|Nørsett|Wanner|1993|loc=§III.1}}; {{harvnb|Quarteroni|Sacco|Saleri|2000}}):
*: <math> \begin{align}
y_n &= y_{n-1} + h f(t_n,y_n) \, </math> — this\qquad\text{(This is the [[backward Euler method]];)}\\
* <math> y_{n+1} &= y_n + \tfrac12frac{1}{2} h \bigleft( f(t_{n+1},y_{n+1}) + f(t_n,y_n) \bigright) </math>, — this\qquad\text{(This is the [[trapezoidal rule (differential equations)|trapezoidal rule]];}\\
* <math> y_{n+2} &= y_{n+1} + h \bigleft( \tfrac5frac{5}{12} f(t_{n+2},y_{n+2}) + \tfrac23frac{2}{3} f(t_{n+1},y_{n+1}) - \tfrac1frac{1}{12} f(t_n,y_n) \bigright); </math>, \\
* <math> y_{n+3} &= y_{n+2} + h \bigleft( \tfrac38frac{3}{8} f(t_{n+3},y_{n+3}) + \tfracfrac{19}{24} f(t_{n+2},y_{n+2}) - \tfrac5frac{5}{24} f(t_{n+1},y_{n+1}) + \tfrac1frac{1}{24} f(t_n,y_n) \bigright). </math>, \\
* <math> \begin{align} y_{n+4} &= y_{n+3} + h \bigleft( &\tfracfrac{251}{720} f(t_{n+4},y_{n+4}) + \tfracfrac{646}{720} f(t_{n+3},y_{n+3}) - \frac{264}{720} f(t_{n+2},y_{n+2}) + \frac{106}{720} f(t_{n+1},y_{n+1}) - \frac{19}{720} f(t_n,y_n) \right) .
\end{align} </math>
&- \tfrac{264}{720} f(t_{n+2},y_{n+2}) + \tfrac{106}{720} f(t_{n+1},y_{n+1}) - \tfrac{19}{720} f(t_n,y_n) \big). \end{align} </math>
The derivation of the Adams–Moulton methods is similar to that of the Adams–Bashforth method; however, the interpolating polynomial uses not only the points ''t''<sub>''n''−1</sub>, … ''t''<sub>''n''−''s''</sub>, as above, but also <math>t_n</math>. The coefficients are given by
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