S= 1-2 + 3-4 + 5-6+... ..+2015-2016(có 2016 số)

=(1-2) + (3-4) + (5-6) +...+(2015-2016) (có 2016:2=1008 nhóm có 2 số)

=-1      +(-1)   +(-1)    +...+(-1)( có 1008 số(-1))

=-1.1008

=-1008

vậy S=-1008

A = 3¹+3²+3³+3⁴+...+3²⁰¹⁵+3²⁰¹⁶

\(S=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{2015.2016}\)

\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{2015}-\frac{1}{2016}\)

\(S=1-\frac{1}{2016}=\frac{2015}{2016}\)

\(S=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-........+\frac{1}{2015}-\frac{1}{2016}\)

\(S=\frac{1}{1}-\left(-\frac{1}{2}+\frac{1}{2}\right)+\left(-\frac{1}{3}+\frac{1}{3}\right)+......+\left(-\frac{1}{2015}+\frac{1}{2015}\right)-\frac{1}{2016}\)

\(S=\frac{1}{1}-\frac{1}{2016}=\frac{2015}{2016}\)

2017/2016

S = 1/1x2 + 1/2x3 + 1/3x4 + ... + 1/2014x2015 + 1/2015x2016

S = 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... +  1/2014 - 1/2015 + 1/2015 - 1/2016

S = 1 - 1/2016

S = 2015

a) 1-2+3-4+...+2016-2017

=(1-2)+(3-4)+......+(2016-2017)

= (-1)+(-1)+......+(-1)

= -1. 1009

= -1009

b) 0-2+4-6+....+2016-2018

=(0-2)+(4-6)+.....+(2016-2018)

=(-2)+(-2)+.....+(-2)

= (-2). 505

=1010

\(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+....+\frac{1}{2^{2015}}+\frac{1}{2^{2016}}\)

\(2A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+.......+\frac{1}{2^{2015}}\)

=>\(2A-A=1+\frac{1}{2}+\frac{1}{2^2}+.......+\frac{1}{2^{2015}}-\left(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...........+\frac{1}{2^{2015}}+\frac{1}{2^{2016}}\right)\)

=>\(A=1-\frac{1}{2^{2016}}\)

=>\(A=\frac{2^{2016}-1}{2^{2016}}\)

Ta có : \(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+.....+\frac{1}{2^{2016}}\)

\(\Rightarrow2A=1+\frac{1}{2}+\frac{1}{2^2}+......+\frac{1}{2^{2016}}\)

\(\Rightarrow2A-A=1-\frac{1}{2^{2016}}\)

\(\Rightarrow A=1-\frac{1}{2^{2016}}\)