Tính tổng S=\(\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
Tính biểu thức sau:
\(S=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
Biến đổi phân số ở dạng tổng quát:
\(\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{3}{3n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{3+n-n}{3n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(=\frac{1}{3}\left[\frac{n+3}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}-\frac{n}{n\left(n+1\right)\left(n+2\right)\left(n+2\right)}\right]\)
=\(\frac{1}{3}\left[\frac{1}{n\left(n+1\right)\left(n+2\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right]\)
Áp dụng kết quả vào bài, ta được:
\(\frac{1}{1.2.3.4}=\frac{1}{3}\left[\frac{1}{1.2.3}-\frac{1}{2.3.4}\right],\frac{1}{2.3.4.5}=\frac{1}{3}\left[\frac{1}{2.3.4}-\frac{1}{3.4.5}\right]\),...
\(\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{1}{3}\left[\frac{1}{n\left(n+1\right)\left(n+2\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right]\)
Cộng từng vế, ta được:
\(S=\frac{1}{3}\left[\frac{1}{1.2.3}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right].\)
\(\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
Đặt A là biểu thức của đề bài.
Ta có: 3/ 1.2.3.4 = 1/ 1.2.3 -1/ 2.3.4
3/ 2.3.4.5 = 1/ 2.3.4 -1/ 3.4.5
3/ n(n+1)(n+2)(n+3) = 1/ n(n+1)(n+2) -1/ (n+1)(n+2)(n+3)
Do đó: 3A = 1/ 1.2.3 -1/ 2.3.4 + 1/ 2.3.4 - 1/ 3.4.5 +...+ 1/ n(n+1)(n+2) - 1/ (n+1)(n+2)(n+3)
3A = 1/ 1.2.3 - 1/ (n+1)(n+2)(n+3)
3A = 1/6 - 1/ (n+1)(n+2)(n+3)
A = 1/18 - 1/ 3(n+1)(n+2)(n+3)
Đó là kết quả rút gọn. Chúc bạn học tốt.
Đặt \(A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
\(\Rightarrow3A=\frac{3}{1.2.3.4}+\frac{3}{2.3.4.5}+\frac{3}{3.4.5.6}+...+\frac{3}{n.\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
\(3A=\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right)}-\frac{1}{\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
\(3A=\frac{1}{1.2.3}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(A=\frac{\frac{1}{1.2.3}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}}{3}\)
B tự làm nốt nhé
Bài này áp dụng công thức:
\(\frac{a}{b.c.d.e}=\frac{1}{b.c.d}-\frac{1}{c.d.e}\)( đk: \(e-b=a\))
hãy k cho tui
tui ko k lại đâu
mại dô!!!!!
THANKS
Tính Sn biết \(S_n=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
Lời giải:
$3S_n=\frac{4-1}{1.2.3.4}+\frac{5-2}{2.3.4.5}+....+\frac{(n+3)-n}{n(n+1)(n+2)(n+3)}$
$=\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+...+\frac{1}{n(n+1)(n+2)}-\frac{1}{(n+1)(n+2)(n+3)}$
$=\frac{1}{1.2.3}-\frac{1}{(n+1)(n+2)(n+3)}$
$\Rightarrow S_n=\frac{1}{1.2.3.3}-\frac{1}{3(n+1)(n+2)(n+3)}$
$\Rightarrow S_n=\frac{1}{18}-\frac{1}{3(n+1)(n+2)(n+3)}$
\(A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(3A=\frac{3}{1.2.3.4}+\frac{3}{2.3.4.5}+...+\frac{3}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(3A=\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(3A=\frac{1}{6}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(3A=\frac{\left(n+1\right)\left(n+2\right)\left(n+3\right)-6}{6\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
=>\(A=\frac{\left(n+1\right)\left(n+2\right)\left(n+3\right)-6}{18\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{n^3+3n^2+3n^2+9n+6-6}{18\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\frac{n^3+6n^2+9n}{18\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
Tính
E=\(\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+......+\frac{ }{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
chỗ phân số thiếu tử thì điền tử bằng 1 nha
dùng sai phân cuối cùng ra:
1- 1/n+3 = n+2 / n+3
\(E=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(=\frac{1}{3}\left(\frac{3}{1.2.3.4}+\frac{3}{2.3.4.5}+...+\frac{3}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right)\)
\(=\frac{1}{3}\left(\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right)\)
\(=\frac{1}{3}\left(\frac{1}{6}-\frac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right)\)
P/S: tham khảo nha
Đến đây bn thu gọn và tính tiếp nhé
Tính nhanh \(B=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
\(B=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right)\left(n+3\right)}\)
\(B=\left(\frac{1}{1.2.3}-\frac{1}{2.3.4}\right)+\left(\frac{1}{2.3.4}-\frac{1}{3.4.5}\right)+...+\left(\frac{1}{n.\left(n+1\right).\left(n+2\right)}-\frac{1}{\left(n+1\right).\left(n+2\right)\left(n+3\right)}\right)\)
\(B=\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right)}-\frac{1}{\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
\(B=\frac{1}{1.2.3}-\frac{1}{\left(n+1\right).\left(n+2\right).\left(n+3\right)}\)
Tính
\(A=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+\frac{1}{3.4.5.6}+...+\frac{1}{19.20.21.22}\)
(Có thể tính hoặc cho công thức với \(\frac{1}{19.20.21.22}=\frac{1}{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}\)
A= \(\frac{1}{3}\left(\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+\frac{1}{3.4.5}-\frac{1}{4.5.6}+...+\frac{1}{19.20.21}-\frac{1}{20.21.22}\right)\)
=\(\frac{1}{3}\left(\frac{1}{6}-\frac{1}{9240}\right)\)
=\(\frac{171}{3080}\)
A=1/1-1/2-1/3+1/2-1/3-1/4-1/5+1/3-1/4-1/5-1/6+...+1/19-1/20-1/21-1/22
A=1/1-1/22
A=21/22
Vậy A=21/22
Tính tổng
\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{n.\left(n+1\right).\left(n+2\right)}\)
Đặt C =\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}\)
\(\Rightarrow2C=\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{n\left(n+1\right)\left(n+2\right)}\)
\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\)
\(=\frac{1}{1.2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\)
\(\Rightarrow C=\left(\frac{1}{1.2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)\div2\)
Tính : \(S_n=\dfrac{1}{1.2.3.4}+\dfrac{1}{2.3.4.5}+...+\dfrac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(S_n=\dfrac{1}{1.2.3.4}+\dfrac{1}{2.3.4.5}+....+\dfrac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
\(S_n=\dfrac{1}{3}\left(\dfrac{1}{1.2.3}-\dfrac{1}{2.3.4}-\dfrac{1}{3.4.5}+....+\dfrac{1}{n\left(n+1\right)\left(n+2\right)}-\dfrac{1}{n\left(n+2\right)\left(n+3\right)}\right)\)\(S_n=\dfrac{1}{3}\left(\dfrac{1}{2.3.4}-\dfrac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right)\)
\(S_n=\dfrac{1}{3}\left(\dfrac{1}{24}-\dfrac{1}{\left(n+1\right)\left(n+2\right)\left(n+3\right)}\right)\)
\(S_n=\dfrac{1}{72}-\dfrac{1}{3\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)