Chứng minh rằng : 1/1.2 + 1/3.4 + 1/4.5 +…+ 1/(2n-1)2n = 1/n+1 + 1/n+2 +… + 1/2n 21/09/2021 Bởi Alaia Chứng minh rằng : 1/1.2 + 1/3.4 + 1/4.5 +…+ 1/(2n-1)2n = 1/n+1 + 1/n+2 +… + 1/2n
Đáp án: Giải thích các bước giải: $\quad\dfrac{1}{1.2}+\dfrac{1}{3.4}+\dfrac{1}{45}+…+\dfrac{1}{(2n-1)(2n)}\\ =1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+…+\dfrac{1}{2n-1}-\dfrac{1}{2n}\\ =1+\dfrac{1}{3}+\dfrac{1}{5}+…+\dfrac{1}{2n-1}-(\dfrac{1}{2}+\dfrac{1}{4}+..+\dfrac{1}{2n})\\ =1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{2n-1}+\dfrac{1}{2n}-2(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+…+\dfrac{1}{2n})\\ =1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{2n-1}+\dfrac{1}{2n}-(1+\dfrac{1}{2}+\dfrac{1}{3}+…+\dfrac{1}{n})\\ =\dfrac{1}{n+1}+\dfrac{1}{n+2}+…+\dfrac{1}{2n}$ Bình luận
1/n-1/n-1=n+1/n(n+1)-n/n(n+1)=n+1-n/n(n+1)=n-n+1/n(n+1)=0+1/n(n+1)=1/n(n+1)
Vậy…..(đpcm)
Đáp án:
Giải thích các bước giải:
$\quad\dfrac{1}{1.2}+\dfrac{1}{3.4}+\dfrac{1}{45}+…+\dfrac{1}{(2n-1)(2n)}\\
=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+…+\dfrac{1}{2n-1}-\dfrac{1}{2n}\\
=1+\dfrac{1}{3}+\dfrac{1}{5}+…+\dfrac{1}{2n-1}-(\dfrac{1}{2}+\dfrac{1}{4}+..+\dfrac{1}{2n})\\
=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{2n-1}+\dfrac{1}{2n}-2(\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{6}+…+\dfrac{1}{2n})\\
=1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{2n-1}+\dfrac{1}{2n}-(1+\dfrac{1}{2}+\dfrac{1}{3}+…+\dfrac{1}{n})\\
=\dfrac{1}{n+1}+\dfrac{1}{n+2}+…+\dfrac{1}{2n}$