Question

Calculati utilizand factorul comun:
a)4+8+12+16+...+400
b)5+10+15+20+...+1000
c)11+22+33+...+1.100

Answer 1

a)S=4+8+12+16+...+400
=4(1+2+3+.....+100)=4[(1+100)*100]/2=4*(101*100)/2=20200

b). S=5(1+2+3+........200)=5*[(1+200)*200]/2=5*(201*200)/2=
100500

c). S= 11(1+2+3+.....100)=11*[(1+100)*100]/2=11*(101*100)/2=55550

Answer 2

$\displaystyle a).4+8+12+16+...+400=4(1+2+3+...+100)= \\ \\ =4 \times \frac{100 (100+1)}{2} =4 \times \frac{100 \times 101}{2} =4 \times \frac{10100}{2} = \\ \\ =4 \times 5050=20200 \\ \\ b).5+10+15+20+...+1000=5(1+2+3+...+200)= \\ \\ =5 \times \frac{200(200+1)}{2} =5 \times \frac{200 \times 201}{2} =5 \times \frac{40200}{2} = \\ \\ =5 \times 20100=100500$

$\displaystyle c).11+22+33+...+1100=11(1+2+3+...+100)= \\ \\ =11 \times \frac{100(100+1)}{2} =11 \times \frac{100 \times 101}{2} =11 \times \frac{10100}{2} = \\ \\ =11 \times 5050=55550$