Đáp án:

-5050.

Giải thích các bước giải:

$1-4+9-16\ +\,.\!.\!.+\ 99^2-100^2\\=1^2-2^2+3^2-4^2\ +\,.\!.\!.+\ 99^2-100^2\\=(1^2+2^2+3^4+4^2\ +\,.\!.\!.+\ 99^2+100^2)-2.(2^2+4^2\ +\,.\!.\!.+100^2)\\=(1^2+2^2+3^4+4^2\ +\,.\!.\!.+\ 99^2+100^2)-2.(2^2.1^2+2^2.2^2\ +\,.\!.\!.+\ 2^2.50^2)\\=(1^2+2^2+3^4+4^2\ +\,.\!.\!.+\ 99^2+100^2)-2.(4.1^2+4.2^2\ +\,.\!.\!.+\ 4.50^2)\\=(\underbrace{1^2+2^2+3^4+4^2\ +\,.\!.\!.+\ 99^2+100^2}_{\large A})-8.(\underbrace{1^2+2^2\ +\,.\!.\!.+\ 50^2}_{\large B})\\A=1^2+2^2+3^4+4^2\ +\,.\!.\!.+\ 99^2+100^2\\\Rightarrow A=1+2.(1+1)+3.(2+1)+4.(3+1)\ +\,.\!.\!.+\ 99.(98+1)+100.(99+1)\\\Rightarrow A=1+1.2+2+2.3+3+3.4+4\ +\,.\!.\!.+\ 98.99+99+99.100+100\\\Rightarrow A=(\underbrace{1.2+2.3+3.4\ +\,.\!.\!.+\ 98.99+99.100}_{\large A_1})+(\underbrace{1+2+3+4\ +\,.\!.\!.+\ 100}_{\large A_2})\\A_1=1.2+2.3+3.4\ +\,.\!.\!.+\ 98.99+99.100\\\Rightarrow 3A_1=1.2.3+2.3.3+3.4.3\ +\,.\!.\!.+\ 98.99.3+99.100.3\\\Rightarrow 3A_1=1.2.3+2.3.(4-1)+3.4.(5-2)\ +\,.\!.\!.+\ 98.99.(100-97)+99.100.(101-98)\\\Rightarrow 3A_1=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4\ +\,.\!.\!.+\ 98.99.100-97.98.99+99.100.101-98.99.100\\\Rightarrow 3A_1=99.100.101\\\Rightarrow A_1=33.100.101\\\Rightarrow A_1=333300\\\ \\A_2=1+2+3+4\ +\,.\!.\!.+\ 100\\\Rightarrow A_2=\dfrac{100.(100+1)}2\\\Rightarrow A_2=50.101\\\Rightarrow A_2=5050\\\ \\\Rightarrow A=A_1+A_2=333300+5050=338350\\\ \\B=1^2+2^2\ +\,.\!.\!.+\ 50^2\\\Rightarrow B=1+2.(1+1)\ +\,.\!.\!.+\ 50.(49+1)\\\Rightarrow B=1+1.2+2\ +\,.\!.\!.+\ 49.50+50\\\Rightarrow B=(\underbrace{1.2\ +\,.\!.\!.+\ 49.50}_{\large B_1})+(\underbrace{1+2\ +\,.\!.\!.+\ 50}_{\large B_2})\\B_1=1.2\ +\,.\!.\!.+\ 49.50\\\Rightarrow 3B_1=1.2.3\ +\,.\!.\!.+\ 49.50.3\\\Rightarrow 3B_1=1.2.3\ +\,.\!.\!.+\ 49.50.(51-48)\\\Rightarrow 3B_1=1.2.3\ +\,.\!.\!.+\ 49.50.51-48.49.50\\\Rightarrow 3B_1=49.50.51\\\Rightarrow B_1=17.49.50\\\Rightarrow B_1=41650\\ \ \\B_2=1+2\ +\,.\!.\!.+\ 50\\\Rightarrow B_2=\dfrac{50.(50+1)}2\\\Rightarrow B_2=25.51\\\Rightarrow B_2=1275\\\ \\\Rightarrow B=B_1+B_2=41650+1275=42925\\\ \\1-4+9-16\ +\,.\!.\!.+\ 99^2-100^2=A-8B=338350-8.42925=-5050$

Giải thích các bước giải:

Ta có :

`1 - 4 + 9 - 16 + ... + 99^2 - 100^2 = 1^2 - 2^2 + 3^2 - 4^2 + 99^2 - 100^2`

Ta lại có :

`1² - 2²  =  1 - 4  =  -3`

`3² - 4²  =  9 - 16  =  -7`

`5² - 6²  =  25 - 36  =  -11`

`7² - 8²  =  49 - 64  =  -15`

`99² - 100² = 9081 - 10000 = -199`

`⇒ -3 + -7 + -11 + -15 + .... + -199 (` Số hạng là `50 )`

⇒ Tổng trên là :

`( -199 + -3 ) . 50 : 2 = -5050`

Vậy tổng trên bằng `-5050`