Đặt \(A=1+2+2^2+2^3+...+2^{100}\\ 2A=2+2^2+2^3+2^4+...+2^{101}\\ 2A-A=2+2^2+2^3+2^4+...+2^{101}-1-2-2^2-2^3-...-2^{100}\\ A=2^{101}-1\)
\(\Rightarrow2^{100}-\left(2^{100}-1\right)=2^{100}-2^{101}+1\)
cmr
100-(1+1/2+1/3+...+1/100)=1/2+2/3+3/4+....+99/100
\(Tính:1+\dfrac{1+2}{2}+\dfrac{1+2+3}{3}+...+\dfrac{1+2+...+100}{100}\)
1. Chứng tỏ rằng tổng 100 số đầu tiên của dãy sau nhỏ hơn 1/4:
1/5; 1/45;1/117;1/221;1/357;...
2.tính A/B biết:
A=1/1.300+1/2.301+1/3.302+...+1/101.400
B=1/1.102+1/2.103+...+1/299.400
3.
Chứng minh rằng; 100-(1+1/2+1/3+...+1/100)=1/2+2/3+...+99/100
4. Tính A/B biết : A=1/2+1/3+...+1/200
B=1/199+2/198+...+199/1
5. Tính: 1-1/2+1/3-1/4+...+1/99-1/100 phần 1/51+1/52+...+1/100
1/3 - 2/3^2 + 3/3^2 - 4/3^4+ ... + 99/3^99 - 100 / 3^100 < 3/16
cm: 1/2^2+1/3^2+1/4^2+...+1/100^2< 1
1: Tính B
\(B=1+\dfrac{1}{2}\cdot\left(1+2\right)+\dfrac{1}{3}\cdot\left(1+2+3\right)+\dfrac{1}{4}\cdot\left(1+2+3+4\right)+...+\dfrac{1}{100}\cdot\left(1+2+3+...+100\right)\)
C=\(\frac{101+100+99+98+...+3+2+1}{101-100+99-98+...+3-2+1}\)
D=\(\frac{3737.43-4343.37}{2+4+6+...+100}\)
Tính
S1=1+2+3+...+999
S2=1-2+3-4+...+99-100+101
S3=1-2-3+4-5-6+...+100
1) tính: 1+ 1/2+ 1/2^2+..........................+ 1/2^99 + 1/2^100 + 1/2^100
