Question

Calculati:
a)4+8+12+16+...+2012
b)7+14+21+28+...+2016
c)3+6+9+12+...+2013
d)5+10+15+20+...+2015

Answer 1

$\displaystyle a).4+8+12+16+...+2012=4(1+2+3+...+503)= \\ \\ =4 \cdot \frac{503(503+1)}{2} =4 \cdot \frac{503 \cdot 504}{2} =4 \cdot \frac{253512}{2} =4 \cdot 126756=507024 \\ \\ b).7+14+21+28+...+2016=7(1+2+3+...+288)= \\ \\ =7 \cdot \frac{288(288+1)}{2} =7 \cdot \frac{288 \cdot 289}{2} =7 \cdot \frac{83232}{2} =7 \cdot 41616=291312 \\ \\ c).3+6+9+12+...+2013=3(1+2+3+...+671)= \\ \\ =3 \cdot \frac{671(671+1)}{2} =3 \cdot \frac{671 \cdot 672}{2} =3 \cdot \frac{450912}{2} =3\cdot 225456=676368$

$\displaystyle d).5+10+15+20+...+2015=5(1+2+3+...+403)= \\ \\ =5 \cdot \frac{403(403+1)}{2} = 5 \cdot \frac{403 \cdot 404}{2} =5 \cdot \frac{162812}{2} =5 \cdot 81406=407030$