Question

calculați:1+1/1+2+1/1+2+3+1/1+2+3+4+5+.....+1/1+2+3+.....+1000

Answer 1

Răspuns:

Explicație pas cu pas:

$1+\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4} +...+\dfrac{1}{1+2+3+...+1000}=\\=1+\dfrac{1}{\frac{(1+2)*2}{2} }+\dfrac{1}{\frac{(1+3)*3}{2} }+\dfrac{1}{\frac{(1+4)*4}{2} }+...+\dfrac{1}{\frac{(1+1000)*1000}{2} }=\dfrac{2}{1*2}+\dfrac{2}{2*3}+\dfrac{2}{3*4}+\\+\dfrac{2}{4*5}+...+\dfrac{2}{1000*1001}=2*(\dfrac{1}{1*2}+\dfrac{1}{2*3}+\dfrac{1}{3*4}+...+\dfrac{1}{1000*1001})=2*(\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{1000}-\dfrac{1}{1001})=$

$=2*(\dfrac{1}{1}-\dfrac{1}{1001})=2*\dfrac{1001-1}{1001}=\dfrac{2000}{1001}$