tinh B= 2+2+2^2+2^3+.....+2^99
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\(B=\dfrac{1}{2}+\dfrac{2}{2^2}+\dfrac{3}{2^3}+.......+\dfrac{99}{2^{99}}+\dfrac{100}{2^{100}}\)
\(\Leftrightarrow2B=1+\dfrac{1}{2^2}+\dfrac{2}{2^3}+\dfrac{3}{2^4}+........+\dfrac{98}{2^{99}}+\dfrac{99}{2^{100}}\)
\(\Leftrightarrow2B-B=\left(1+\dfrac{1}{2^2}+\dfrac{2}{2^3}+........+\dfrac{99}{2^{100}}\right)-\left(\dfrac{1}{2}+\dfrac{2}{2^2}+......+\dfrac{100}{2^{100}}\right)\)
\(\Leftrightarrow B=\dfrac{1}{2}+\dfrac{1}{2^2}+..........+\dfrac{1}{2^{100}}-\dfrac{100}{2^{100}}\)
Đặt :
\(A=\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{100}}\)
\(\Leftrightarrow2A=1+\dfrac{1}{2}+\dfrac{1}{2^2}+........+\dfrac{1}{2^{99}}\)
\(\Leftrightarrow2A-A=\left(1+\dfrac{1}{2}+......+\dfrac{1}{2^{99}}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+.....+\dfrac{1}{2^{100}}\right)\)
\(\Leftrightarrow A=1-\dfrac{1}{2^{100}}\)
\(\Leftrightarrow B=1-\dfrac{1}{2^{100}}-\dfrac{100}{2^{100}}\)
\(\Leftrightarrow B=\dfrac{2^{100}-101}{2^{100}}\)
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![](https://rs.olm.vn/images/avt/0.png?1311)
Bạn giỏi bạn làm đi đã ngu zồi thích tỏ ra minh ngu hơn. Bạn sợ bạn nếu ko nói câu đấy người ta tưởng bạn khôn chắc
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1 - 2 - 3 + 4 + 5 - 6 - 7 + 8+ ... + 1993 - 1994
= ( 1 - 2 - 3 + 4 ) = ( 5 - 6 - 7 + 8 ) + ... + 1993 - 1994
= 0 + 0 + ... + 1993 - 1994
= 0 + ( -1 ) = -1
b) ta có 1^2+2^2+...+n^2 = n(n+1)(2n+1)/6
=>2^2+4^2+...+(2n)^2= 2^2(1^2+2^2+...+n^2)= 2n(n+1)(2n+1)/3
và 1^2+2^2+...+(2n+1)^2=(2n+1)(2n+2)(4n+3)/...
=>1^2+3^2+5^2+...+(2n+1)^2 = (2n+1)(2n+2)(4n+3)/6 - 2n(n+1)(2n+1)/3 = (2n+1)(n+1)(2n+3)/3
=>1^2-2^2+3^2-4^2+..... -(2n)^2+(2n+1)^2 = (2n+1)(n+1)(2n+3)/3 - 2n(n+1)(2n+1)/3 = (n+1)(2n+1)
do đó ta có khi n = 100 thì
1^2-2^2+3^2-4^2.....+99^2-100^2+101^2 = (100+1)*(2*100+1)=201*101
Mình cũng không chắc câu b cho lắm
![](https://rs.olm.vn/images/avt/0.png?1311)
\(1\times2+2\times3.........+100\times2\)
\(=\left(1+100\right)+\left(2+99\right)+...+\left(50+51\right)\times2\)(50 cặp)
\(=\left(101+101+....+101\right)\times2\)(50 số 101)
\(=101\times50\times2\)
\(=10100\)
k mình nha
B = 2+ 22+23+...+299
=> 2B=22+23+24+...+2100
=> 2B-B=2100-2
=> B = 2100-2