等和的二個等差級數

1+2=3

4+5+6=(4+3)+(5+3)=7+8

9+10+11+12=(9+4)+(10+4)+(11+4)=13+14+15

16+17+18+19+20=(16+5)+(17+5)+(18+5)+(19+5)=21+22+23+24

 

如果 n是自然數,有(n+1)項且公差是1的等差級數 n2+ ....... +(n2+n) 有n項且公差是1的等差級數 (n2+n+1)+ ....... + (n2+2n),則兩個等差級數和相等

說明

n2+ (n2+1)+ (n2+2)....... + (n2+n-1)+(n2+n)=

n2+ (n2+1)+ (n2+2)....... + (n2+n-1)+n (n+1) =

[n2+(n+1)]+[(n2+1)+(n+1)]+....+ [(n2+n-1)+(n+1)]=

 [(n+1)2-n]+ [(n+1)2-n+1]+ ....... +(n2+2n)


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