tính A=1/2!+2/3!+3/4!+...+2021/2022!

2 câu trả lời

12!+23!+34!+....+20212022!

= ( 2 - 1 )/( 2! ) + ( 3 - 1 )/( 3! ) + ( 4 - 1 )/( 4! ) + .... + ( 2022 - 1 )/( 2022! )

= 2/( 2! ) - 1/( 2! ) + 3/( 3! ) - 1/( 3! ) + 4/( 4! ) - 1/( 4! ) + .... + 2022/( 2022! ) - 1/( 2022! )

= 1/( 1! ) - 1/( 2! ) + 1/( 2! ) - 1/( 3! ) + 1/( 3! ) - 1/( 4! ) + .... + 1/( 2021! ) - 1/( 2022! )

= 1/( 1! ) - 1/( 2022! )

= 1 - 1/( 2022! )

\quadA=1/(2!)+2/(3!)+3/(4!)+...+2021/(2022!)

\quad*** \text{Ta có:}

\quad\bullet\quad1/(2!)=(2-1)/(2!)=2/(2!)-1/(2!)=1/(1!)-1/(2!)

\quad\bullet\quad2/(3!)=(3-1)/(3!)=3/(3!)-1/(3!)=1/(2!)-1/(3!)

\quad\bullet\quad3/(4!)=(4-1)/(4!)=4/(4!)-1/(4!)=1/(3!)-1/(4!)

\quad\bullet\quad2021/(2022!)=(2022-1)/(2022!)=(2022)/(2022!)-1/(2022!)=1/(2021!)-1/(2022!)

\quad\boxed{\Xi}\quad \text{Cộng vế theo vế các đăng thức, ta được:}

\quadA=1/(1!)-1/(2!)+1/(2!)-1/(3!)+1/(3!)-1/(4!)+1/(2021!)-1/(2022!)

\quad->A=1/(1!)-1/(2022!)

\quad->A=1-1/(2022!)

\quad\text{Vậy,} A=1-1/(2022!).