\(A=1.2+2.3+...+25.26\)
\(\Rightarrow3A=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+25.26.\left(27-24\right)\)
\(=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+25.26.27-24.25.26\)
\(\Rightarrow3A=25.26.27\)
\(\Rightarrow A=25.26.9\)
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