Just an Ordinary Arithmetic

$\begin{aligned} S & = \frac 1{1\times 4\times 7} + \frac 1{4\times 7\times 10} + \frac 1{7\times 10\times 13} + \cdots + \frac 1{19\times 22\times 25} & \small \color{#3D99F6} \text{By partial fraction decomposition} \\ & = \frac 1{18} \left(\frac 11 \color{#3D99F6} - \frac 24 \color{#D61F06} + \frac 17 \color{#3D99F6} + \frac 14 \color{#D61F06} - \frac 27 \color{#3D99F6} + \frac 1{10} \color{#D61F06} + \frac 17 \color{#3D99F6} - \frac 2{10} \color{#D61F06} + \frac 1{13} \color{#3D99F6}+ \cdots \color{#D61F06}+ \frac 1{19} {\color{#3D99F6} - \frac 2{22}} + \frac 1{25} \right) \\ & = \frac 1{18}\left(1 - \frac 14 - \frac 1{22} + \frac 1{25} \right) = \frac {91}{2200} \end{aligned}$

Therefore, $a+b = 91+2200 = \boxed{2291}$ .

$\frac 1{1 \times 4 \times 7} + \frac 1{4 \times 7 \times 10} + \frac 1{7 \times 10 \times 13} + \cdots + \frac 1{19 \times 22 \times 25}$

$=\frac16\left((\frac1{1\times4}-\frac1{4\times7})+(\frac1{4\times7}-\frac1{7\times10})+...+(\frac1{19\times22}-\frac1{22\times25})\right)$

$=\frac1{1\times4\times6}-\frac1{22\times25\times6}$

$=\frac{91}{2200}$