Tính P biết :
\(p=1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+.....+\frac{1}{16}\left(1+2+...+16\right)\)
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Những câu hỏi liên quan
3 tháng 12 2016
Đặt \(A=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{16}\left(1+2+3+...+16\right)\)
\(A=1+\frac{1+2}{2}+\frac{1+2+3}{3}+\frac{1+2+3+4}{4}+...+\frac{1+2+3+...+16}{16}\)
\(A=1+\frac{2\left(2+1\right):2}{2}+\frac{3\left(3+1\right):2}{3}+\frac{4\left(4+1\right):2}{4}+...+\frac{16\left(16+1\right):2}{16}\)
\(A=1+\frac{2+1}{2}+\frac{3+1}{2}+\frac{4+1}{2}+...+\frac{16+1}{2}\)
\(A=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{17}{2}\)
\(A=\frac{2+3+4+5+...+17}{2}\)
\(A=\frac{152}{2}\)
\(A=76\)
1 tháng 11 2016
\(=1+\frac{3}{2}+2+\frac{5}{2}+3+\frac{7}{2}+...+8+\frac{17}{2}\)
\(=\left(1+2+...+8\right)+\left(\frac{3}{2}+\frac{5}{2}+...+\frac{17}{2}\right)=36+\frac{80}{2}=36+40=76\)
1 tháng 11 2016
\(B=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{16}.\left(1+2+3+...+16\right)\)
\(B=1+\frac{1}{2}.\frac{\left(1+2\right).2}{2}+\frac{1}{3}.\frac{\left(1+3\right).3}{2}+\frac{1}{4}.\frac{\left(1+4\right).4}{2}+...+\frac{1}{16}.\frac{\left(1+16\right).16}{2}\)
\(B=1+\frac{1}{2}.\frac{3.2}{2}+\frac{1}{3}.\frac{4.3}{2}+\frac{1}{4}.\frac{5.4}{2}+...+\frac{1}{16}.\frac{17.16}{2}\)
\(B=1+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{17}{2}\)
\(B=\frac{1}{2}.\left(2+3+4+5+...+17\right)\)
\(B=\frac{1}{2}.\frac{\left(2+17\right).16}{2}=19.4=76\)
4 tháng 5 2016
A=1+\(\frac{1+2}{2}\)+\(\frac{1+2+3}{3}\)+...+\(\frac{1+2+3+...+16}{16}\)
A=\(\frac{2}{2}\)+\(\frac{3}{2}\)+\(\frac{4}{2}\)+...+\(\frac{17}{2}\)
A=\(\frac{2+3+4+...+17}{2}\)
A=76(đề thi HSG huyện tui có tui làm zậy mà cũng có điểm tuyệt đối)
MC
4 tháng 5 2016
\(A=1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)+....+\frac{1}{16}.\left(1+2+3+....+16\right)\)
\(A=1+\frac{1}{2}\cdot\frac{2.3}{2}+\frac{1}{3}\cdot\frac{3.4}{2}+\frac{1}{4}\cdot\frac{4.5}{2}+.....+\frac{1}{16}\cdot\frac{16.17}{2}\)
\(A=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+....+\frac{17}{2}\)
\(A=\frac{\left(2+3+4+.....+17\right)}{2}=\frac{\left(2+17\right).16}{2}=\frac{152}{2}=76\)
13 tháng 9 2018
A = 1 + 1/2.(1+2) + 1/3.(1+2+3) + 1/4.(1+2+3+4) + ...+ 1/16.(1+2+....+16)
A = 1 + 1/2.3 + 1/3.6 + 1/4.10 + ...+ 1/16.136
A = 1 + 3/2 + 4/2 + 5/2 + ....+ 17/2
A = 1 + (3+4+5+...+17)/2
A = 1 + 150/2
A = 1 + 75
A = 76
13 tháng 9 2018
A = 1 + 1/2.(1+2) + 1/3.(1+2+3) + 1/4.(1+2+3+4) + ...+ 1/16.(1+2+....+16)
A = 1 + 1/2.3 + 1/3.6 + 1/4.10 + ...+ 1/16.136
A = 1 + 3/2 + 4/2 + 5/2 + ....+ 17/2
A = 1 + (3+4+5+...+17)/2
A = 1 + 150/2
A = 1 + 75
A = 76
MC
5 tháng 5 2016
\(A=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+....+\frac{1}{16}\left(1+2+3+....+16\right)\)
\(A=1+\frac{1}{2}\cdot\frac{2.3}{2}+\frac{1}{3}\cdot\frac{3.4}{2}+....+\frac{1}{16}\cdot\frac{16.17}{2}\)
\(A=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+.....+\frac{17}{2}\)
\(A=\frac{\left(2+3+4+....+17\right)}{2}=\frac{\left(2+17\right).\left(17-2+1\right):2}{2}=\frac{152}{2}=76\)
CG
5 tháng 5 2016
Ta có: \(A = 1+{1+2\over 2} + {1+2+3\over 3} +...+{1+2+...+ 16\over 16}\)
Xét: \(S_n = 1+2+3+...+n =\frac{n(n+1)}{n} (n \in N^*)\)
=> \({S_n\over n} = {(n+1)\over 2}\)
Thay vào biểu thức A ta có:
\(A=1 + {3\over 2} + {4\over 2} + ... + {17\over 2}\)
\(A={(2+3+4+...+17)\over 2}\)
\(A={(17+2)[(17-2+1):2]\over 2} = {152\over2}=76\)
7 tháng 2 2017
\(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+\frac{1}{4}\left(1+2+3+4\right)+...+\frac{1}{16}\left(1+2+3+...+16\right)\)
\(=1+\frac{1}{2}.\frac{2.3}{2}+\frac{1}{3}.\frac{3.4}{2}+\frac{1}{4}.\frac{4.5}{2}+...+\frac{1}{16}.\frac{16.17}{2}\)
\(=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+\frac{5}{2}+...+\frac{17}{2}\)
\(=\frac{\frac{17.18}{2}-1}{2}=76\)
15 tháng 5 2015
\(A=1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)+...+\frac{1}{16}.\left(1+2+...+16\right)\)
\(=1+\frac{1}{2}.2.3:2+\frac{1}{3}.3.4:2+...+\frac{1}{16}.16.17:2=1+\frac{3}{2}+\frac{4}{2}+...+\frac{17}{2}=\frac{2+3+4+...+17}{2}=\frac{152}{2}=76\)
Ta có công thức : 1 + 2 + 3 + ... + n = \(\frac{n\left(n+1\right)}{2}\)
Do đó
P = \(1+\frac{1+2}{2}+\frac{1+2+3}{3}+...+\frac{1+2+3+...+16}{16}\)
\(P=1+\frac{2.3}{2.2}+\frac{3.4}{2.3}+\frac{4.5}{2.4}+...+\frac{16.17}{2.16}\)
\(P=1+\frac{1}{2}\left(3+4+5+...+17\right)\)
\(P=1+\frac{1}{2}.\frac{\left(17-3+1\right)\left(3+17\right)}{2}=76\)
Xét thừa số tổng quát:
\(\frac{1+2+3+...+n}{n}=\frac{n\left(n+1\right)}{2n}=\frac{n+1}{2}\)
Thay vào bài toán:
\(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{16}\left(1+2+3+...+16\right)\)
\(=1+\frac{2+1}{2}+\frac{3+1}{2}+...+\frac{16+1}{2}=\frac{2+3+...+17}{2}=76\)