Question

1+ 1/1+2 +1/1+2+3 +1/1+2+3+4 +....+1/1+2+3+...+x=400/201​

Answer 1

Răspuns:

Explicație pas cu pas:

1+ 1/1+2 +1/1+2+3 +1/1+2+3+4 +....+1/1+2+3+...+x=    (folosim sume Gauss)

1+1/2·3/2+1/3·4/2+...+1/x(x+1)/2=

1+2/2·3+2/3·4+.....+2/x(x+1)+1-1=

2[1+1/2·3+1/3·4+...+1/x(x+1)]-1=

1/2·3=1/2-1/3                      folosim formula 1/x(x+1)=1/x-1/(x+1)

1/3·4=1/3-1/4

1/(x-1)x=1/(x-1)-1/x

1/x(x+1)=1/x-1/(x+1)

.............................

2+ 1-1/(x+1)-1=400/201 ⇒2x/(x+1)=400/201     x/(x+1)=200/201

201x=200x+200      ⇒x=200