Đáp án:

$1^2+2^2+3^2\ +\,.\!.\!.+\ 100^2=338350$.

Giải thích các bước giải:

$\begin{array}{l}1^2+2^2+3^2\ +\,.\!.\!.+\ 100^2\\=1+2.2+3.3\ +\,.\!.\!.+\ 100.100\\=1+2.(1+1)+3.(2+1)\ +\,.\!.\!.+\ 100.(99+1)\\=1+1.2+2+2.3+3\ +\,.\!.\!.+\ 99.100+100\\=\bigg(\underbrace{1.2+2.3\ +\,.\!.\!.+\ 99.100}_{\large A}\bigg)\!+\!\bigg(\underbrace{1+2+3\ +\,.\!.\!.+\ 100}_{\large A'}\bigg)\\\ \\A=1.2+2.3\ +\,.\!.\!.+\ 99.100\\\Rightarrow 3A=1.2.3+2.3.3\ +\,.\!.\!.+\ 99.100.3\\\Rightarrow 3A=1.2.3+2.3.(4-1)\ +\,.\!.\!.+\ 99.100.(101-98)\\\Rightarrow 3A=1.2.3+2.3.4-1.2.3\ +\,.\!.\!.+\ 99.100.101-98.99.100\\\Rightarrow 3A=(1.2.3+2.3.4\ +\,.\!.\!.+\ 99.100.101)-(1.2.3\ +\,.\!.\!.+\ 98.99.100)\\\Rightarrow 3A=99.100.101\\\Rightarrow A=\dfrac{99.100.101}3\\\Rightarrow A=33.100.101\\\Rightarrow A=333300\\\ \\\end{array}$

$A'=1+2+3\ +\,.\!.\!.+\ 100$ (có 100 số).

$\Rightarrow A'=(100+1)+(2+99)+(3+98)\ +\,.\!.\!.$ 
$\Rightarrow A'=101+101+101\ +\,.\!.\!.$ (có $\dfrac{100}2=50$ số).
$\Rightarrow A'=101.50$

$\Rightarrow A'=5050$

$\Rightarrow 1^2+2^2+3^2\ +\,.\!.\!.+\ 100^2=A+A'$

$=333300+5050$

$=338350$.

Vậy $1^2+2^2+3^2\ +\,.\!.\!.+\ 100^2=338350$.

1²+2² +3²+4²+...+99²

^{Ta gọi tên biểu thức là A.}

^{Ta có:}

^A=1²+2² +3²+4²+...+99²

^{A=1.1+2.2+3.3+4.4+...+99.99}

^{A=1.(2-1)+2.(3-1)+3.(4-1)+4.(5-1)+...+99.(100-1)}

^{A=1.2-1.1+2.3-2.1+3.4-3.1+4.5-4.1+...+99.100-99.1}

^{A=1.2+2.3+3.4+4.5+...+99.100-1.1-2.1-3.1-4.1-...-99.1}

^{A=(1.2+2.3+3.4+4.5+...+99.100)-(1.1+2.1+3.1+4.1+...+99.1)}

A=(1.2+2.3+3.4+4.5+...+99.100)-(1+2+3+4+...+99)

Xét:

B=1.2+2.3+3.4+4.5+...+99.100

3B=3.(1.2+2.3+3.4+4.5+...+99.100)

3B=3.1.2+3.2.3+3.3.4+3.4.5+...+3.99.100

3B=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+4.5.6-3.4.5+...+99.100.101-98.99.100

3B=99.100.101-0.1.2

3B=99.101.100-0

3B=9999.100

3B=999900

B=999900:3

B=333300

Xét:

C=1+2+3+4+...+99

C=99+98+97+96+...+1

⇒C+C=(1+99)+(2+98)+(3+97)+(4+96)+...+(99+1)

2C=100+100+100+100+...+100

Có số số 100 là :

(99-1):1+1=99(số)

2C=99.100

2C=9900

C=9900:2

C=4950

A=(1.2+2.3+3.4+4.5+...+99.100)-(1+2+3+4+...+99)

A= B - C

A= 333300 - 4950

A= 328350

Vậy giá trị của 1²+2² +3²+4²+...+99² là 328350