Question

Aratati ca: 1+3+5+...+(2p-1)=p la puterea 2 unde p E N*

Answer 1

S=1+3+5+7+....(2p-1);;;S=(2p-1)+(2p-3)+.......1 => 2S=2p+2p+....2p ( vor fi p termeni. => 2S=2pxp => S=2p la putetea 2/2 deci S= p la puterea2

Answer 2

1+3+5+.....+(2p-1)=(2*0+1)+(2*1+1)+(2*2+1)+.....+[2*(p-1)+1]=
Aranjam convenabili termenii:
1+1+1+.......+1+2*0+2*1+2*2+.......+2*(p-1)=n+2*[1+2+3+....+(p-1)]=
Se stie ca 1+2+3+.......+n=n*(n+1):2
p+2*(p-1)*p:2(*2 cu :2 se simplifica)=p+(p-1)*p=p*[1+(p-1)]=
p*(p-1+1)=p*p=p²