Question

5. Resuelve:
5² +10² +15²+20² + ... +45²
a) 6025
b) 6125
c) 7025
d) 7050
e) 7125​

Answer 1

Respuesta:

E. 7125

Explicación paso a paso:

5²+10²+15²+20²+25²+30²+35²+40²+45²

25+100+225+400+625+900+1225+1600+2025 =7125

Answer 2

Respuesta:

7125

Explicación paso a paso:

5² +10² +15²+20² + ... +45²

Aplicamos un artificio matemático

x = 5

Sustituimos toda la ecuación x = 5

$x^2 + (x+5)^2+ (x+10)^2 + (x+15)^2 + (x+20)^2 + (x+25)^2 + (x+30)^2 + (x+35) ^2 + (x+40)^2$

$\mathrm{Aplicar\:la\:formula\:del\:binomio\:al\:cuadrado}:\quad \left(a+b\right)^2=a^2+2ab+b^2$

$\\\\x^2+x^2+10x+25+\left(x+10\right)^2+\left(x+15\right)^2+\left(x+20\right)^2+\left(x+25\right)^2+\left(x+30\right)^2+\left(x+35\right)^2+\left(x+40\right)^2\\\\x^2+x^2+10x+25+x^2+20x+100+\left(x+15\right)^2+\left(x+20\right)^2+\left(x+25\right)^2+\left(x+30\right)^2+\left(x+35\right)^2+\left(x+40\right)^2\\\\x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+\left(x+20\right)^2+\left(x+25\right)^2+\left(x+30\right)^2+\left(x+35\right)^2+\left(x+40\right)^2$

$x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+x^2+40x+400+\left(x+25\right)^2+\left(x+30\right)^2+\left(x+35\right)^2+\left(x+40\right)^2\\\\x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+x^2+40x+400+x^2+50x+625+\left(x+30\right)^2+\left(x+35\right)^2+\left(x+40\right)^2\\\\x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+x^2+40x+400+x^2+50x+625+x^2+60x+900+\left(x+35\right)^2+\left(x+40\right)^2$

$x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+x^2+40x+400+x^2+50x+625+x^2+60x+900+x^2+70x+1225+\left(x+40\right)^2\\\\x^2+x^2+10x+25+x^2+20x+100+x^2+30x+225+x^2+40x+400+x^2+50x+625+x^2+60x+900+x^2+70x+1225+x^2+80x+1600$

$x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+10x+20x+30x+40x+50x+60x+70x+80x+25+100+225+400+625+900+1225+1600\\\\\mathrm{Sumar\:elementos\:similares:}\:x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2+x^2=9x^2\\\\\mathrm{Sumar\:elementos\:similares:}\:10x+20x+30x+40x+50x+60x+70x+80x=360x\\\\\mathrm{Sumar:}\:25+100+225+400+625+900+1225+1600=5100$

$9x^2+360x+5100$

Remplazar x = 5 en $9x^2+360x+5100$

$9x^2+360x+5100$

$9(5)^2+360(5)+5100\\\\9(25)+1800+5100\\\\225+1800+5100\\\\7125$