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4B=1.2.3.4+2.3.4.4+...+(n-1)n(n+1).4
=1.2.3.4-0.1.2.3+2.3.4.5-1.2.3.4+...+(n-1)n(n+1)(n+2)-[(n-2)(n-1)n(n+1)]
=(n-1)n(n+1)(n+2)-0.1.2.3=(n-1)n(n+1)(n+2)
=>B=(n-1)n(n+1)(n+2)/4
k nha
Ta có : B=1.2.3.4+2.3.4.4+....+(n-1)n(n+1).4
= 1.2.3.4 + 2.3.4.(5-1) + 3.4.5.(6-2) + ... + (n-1)n(n+1)[(n+2)-(n-2)]
=1.2.3.4 +2.3.4.5 - 1.2.3.4 + 3.4.5.6 - 2.3.4.5 + .... + (n-1)n(n+1).(n+2) - (n-2).(n-1).n(n+1)
= ( 1.2.3.4 - 1.2.3.4 ) + ( 2.3.4.5 - 2.3.4.5 ) + .... + ( n-1).n.(n+1).(n+2)
= 0 + 0 + 0 + ... + ( n-1).n.(n+1).(n+2)
= ( n-1).n.(n+1).(n+2)
Vậy B = ( n-1).n.(n+1).(n+2)
\(B=1.2.3+2.3.4+...+\left(n-1\right).n.\left(n+1\right)\)
\(4B=1.2.3.4+2.3.4.\left(5-1\right)+...+\left(n-1\right).n.\left(n+1\right)\left[\left(n+2\right)-\left(n-2\right)\right]\)
\(4B=1.2.3.4+2.3.4.5-1.2.3.4+...+\left(n-1\right).n.\left(n+1\right)\left(n+2\right)-\left(n-2\right)\left(n-1\right).n.\left(n+1\right)\)
\(4B=\left(n-1\right).n.\left(n+1\right)\left(n+2\right)\)
\(B=\frac{\left(n-1\right).n.\left(n+1\right)\left(n+2\right)}{4}\)
Tham khảo nhé~
Ta có: \(B=1.2.3+2.3.4+...+\left(n-1\right).n.\left(n+1\right)\)
\(\Leftrightarrow4B=4.\left[1.2.3+2.3.4+...+\left(n-1\right).n.\left(n+1\right)\right]\)
\(\Leftrightarrow4B=1.2.3.4+2.3.4.4+...+\left(n-1\right).n.\left(n+1\right).4\)
\(\Leftrightarrow4B=1.2.3.4+2.3.4\left(5-1\right)+...+\left(n-1\right)n.\left(n+1\right).\left[\left(n+2\right)-\left(n-2\right)\right]\)
\(\Leftrightarrow4B=1.2.3.4+2.3.4.5-1.2.3.4+...+\left(n-1\right).n.\left(n+1\right).\left(n+2\right)-\left(n-2\right).\)\(\left(n-1\right).n.\left(n+1\right)\)
\(\Leftrightarrow4B=\left(n-1\right).n.\left(n+1\right).\left(n+2\right)\)
\(\Leftrightarrow B=\left(n-1\right).n.\left(n+1\right).\left(n+2\right)\div4\)
Vậy \(B=\left(n-1\right).n.\left(n+1\right).\left(n+2\right)\div4\)
S = 1.2.3 + 2.3.4 +..+ (n-1).n.(n+1)
4S = 1.2.3.4 + 2.3.4.4 + 3.4.5.4 +..+ (n-1)n(n+1).4
ghi dọc cho dễ nhìn:
(k-1)k(k+1).4 = (k-1)k(k+1)[(k+2) - (k-2)] = (k-1)k(k+1)(k+2) - (k-2)(k-1)k(k+1)
ad cho k chạy từ 2 đến n ta có:
1.2.3.4 = 1.2.3.4
2.3.4.4 = 2.3.4.5 - 1.2.3.4
3.4.5.4 = 3.4.5.6 - 2.3.4.5
...
(n-2)(n-1)n.4 = (n-2)(n-1)n(n+1) - (n-3)(n-2)(n-1)n
(n-1)n(n+1).4 = (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1)
+ + cộng lại vế theo vế + + (chú ý cơ chế rút gọn)
4S = (n-1)n(n+1)(n+2)
=> S = (n-1)n(n+1)(n+2)/4
\(\frac{1}{16}<\left(\frac{1}{2}\right)^n<\frac{1}{4}\)
\(\left(\frac{1}{2}\right)^4<\left(\frac{1}{2}\right)^n<\left(\frac{1}{2}\right)^2\)
2 < n < 4 => n = 3
B = 1.2.3 + 2.3.4 + ... + (n - 1)n(n + 1)
4B = 1.2.3.4 + 2.3.4.4 + ... + (n - 1)n(n + 1).4
4B = 1.2.3.4 + 2.3.4.(5 - 1) + 3.4.5.(6 - 2) + .... + (n - 1).n.(n + 1).[(n + 2) - (n - 2)]
4B = 1.2.3.4 + 2.3.4.5 - 1.2.3.4 + 3.4.5.6 - 2.3.4.5 + ... + (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1)
4B = (n-1)n(n+1)(n+2)
B = (n-1)n(n+1)(n+2) : 4
Ta có : 4B =4 . ( 1.2.3 + 2.3.4 + ...+ (n - 1 )n( n + 1 )
<=> 4B = 1.2.3 .( 4 - 0 ) + 2.3.4 .( 5- 1 ) + ... + ( n - 1 ) n ( n + 1 ) [ ( n + 2 ) - ( n - 2 ) ]
<=> 4B = 1 . 2 . 3 . 4 +2 . 3. 4 .5 -1.2.3 .4 + ... + ( n- 1 ) n ( n + 1 ) ( n + 2 )- ( n-1)( n+1).n/( n- 2 )
<=> 4B = ( n- 1 ).( n+1 ).n.( n + 2 )
<=> B = \(\frac{\left(n-1\right)\left(n+1\right)n\left(n+2\right)}{4}\)
Vậy B = \(\frac{\left(n-1\right)\left(n+1\right)n\left(n+2\right)}{4}\)