Let's calculate: $\sum _{n=1}^{12}{n}^2$

ⅰ $\sum _{n=1}^{12}{n}^2$ = $1^2$$+2^2$$+3^2$$+4^2$$+5^2$$+6^2$$+7^2$$+8^2$$+9^2$$+{10}^2$$+{11}^2$$+{12}^2$

$1^2$$+2^2$$+3^2$$+4^2$$+5^2$$+6^2$$+7^2$$+8^2$$+9^2$$+{10}^2$$+{11}^2$$+{12}^2$

ⓘ Calculate powers before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product: $2^2$ = $2·2$ = $4$, $9^2$ = $9·9$ = $81$■Powers are computed before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product: $3^2$ = $3·3$ = $9$, $8^2$ = $8·8$ = $64$, ${10}^2$ = $10·10$ = $100$, ${11}^2$ = $11·11$ = $121$, ${12}^2$ = $12·12$ = $144$■Calculate powers before multiplications, additions and subtractions. With a positive integer exponent, a power corresponds to a repeated product: $4^2$ = $4·4$ = $16$, $5^2$ = $5·5$ = $25$■Powers are computed before multiplications, additions and subtractions. A power with a positive integer exponent is a repeated product: $6^2$ = $6·6$ = $36$, $7^2$ = $7·7$ = $49$

$1$$+4$$+9$$+16$$+25$$+36$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $1$ and move forward $4$ on the number line: $1+4$ = $5$

$5$$+9$$+16$$+25$$+36$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $5$ and move forward $9$ on the number line: $5+9$ = $14$

$14$$+16$$+25$$+36$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $14$ and move forward $16$ on the number line: $14+16$ = $30$

$30$$+25$$+36$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $30$ and move forward $25$ on the number line: $30+25$ = $55$

$55$$+36$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $55$ and move forward $36$ on the number line: $55+36$ = $91$

$91$$+49$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $91$ and move forward $49$ on the number line: $91+49$ = $140$

$140$$+64$$+81$$+100$$+121$$+144$

ⓘ Start from $140$ and move forward $64$ on the number line: $140+64$ = $204$

$204$$+81$$+100$$+121$$+144$

ⓘ Start from $204$ and move forward $81$ on the number line: $204+81$ = $285$

$285$$+100$$+121$$+144$

ⓘ Start from $285$ and move forward $100$ on the number line: $285+100$ = $385$

$385+121+144$

ⓘ Start from $385$ and move forward $121$ on the number line: $385+121$ = $506$

$506+144$

ⓘ Start from $506$ and move forward $144$ on the number line: $506+144$ = $650$

▶Result:   $650$