Đáp án:
$25164150$
Giải thích các bước giải:
$1.2^2+2.3^2+3.4^2\ +\,.\!.\!.+\ 99.100^2\\=1.2.2+2.3.3+3.4.4\ +\,.\!.\!.+\ 99.100.100\\=1.2.(3-1)+2.3.(4-1)+3.4.(5-1)\ +\,.\!.\!.+\ 99.100.(101-1)\\=1.2.3-1.2+2.3.4-2.3+3.4.5-3.4\ +\,.\!.\!.+\ 99.100.101-99.100\\=\bigg(\underbrace{1.2.3+2.3.4+3.4.5\ +\,.\!.\!.+\ 99.100.101}_{\large A}\bigg)-\bigg(\underbrace{1.2+2.3+3.4\ +\,.\!.\!.+\ 99.100}_{\large A'}\bigg)\\\Rightarrow A=1.2.3+2.3.4+3.4.5\ +\,.\!.\!.+\ 99.100.101\\\Rightarrow 4A=1.2.3.4+2.3.4.4+3.4.5.4\ +\,.\!.\!.+\ 99.100.101.4\\\Rightarrow4A=1.2.3.4+2.3.4.(5-1)+3.4.5.(6-2)\ +\,.\!.\!.+\ 99.100.101.(102-98)\\\Rightarrow 4A=1.2.3.4+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5\ +\,.\!.\!.+\ 99.100.101.102-98.99.100.101\\\Rightarrow 4A=(1.2.3.4+2.3.4.5+3.4.5.6\ +\,.\!.\!.+\ 99.100.101.102)-(1.2.3.4+2.3.4.5+3.4.5.6\ +\,.\!.\!.+\ 98.99.100.101)\\\Rightarrow 4A=99.100.101.102\\\Rightarrow A=25.99.101.102\\\Rightarrow A=25497450\\\Rightarrow A'=1.2+2.3+3.4\ +\,.\!.\!.+\ 99.100\\\Rightarrow3A'=1.2.3+2.3.3+3.4.3\ +\,.\!.\!.+\ 99.100.3\\\Rightarrow 3A'=1.2.3+2.3.(4-1)+3.4.(5-2)\ +\,.\!.\!.+\ 99.100.(101-98)\\\Rightarrow 3A'=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4\ +\,.\!.\!.+\ 99.100.101-98.99.100\\\Rightarrow 3A'=(1.2.3+2.3.4+3.4.5\ +\,.\!.\!.+\ 99.100.101)-(1.2.3+2.3.4+3.4.5\ +\,.\!.\!.+\ 98.99.100)\\\Rightarrow 3A'=99.100.101\\\Rightarrow A'=33.100.101\\\Rightarrow A'=333300\\=25497450-333300\\=25164150$
