a, - { -(2016 +2015) - [ - (2016 - 2015) - (2016+2015) ] }

= -{-(2016+2015)-[-0-0]}

= -{-4031-0-0}

=-4031

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

=>A<\(\frac{2016^{2016}+2-2}{2016^{2016}-1-2}\)=\(\frac{2016^{2016}}{2016^{2016}-3}\)=B

=>A<B(công thức nếu \(\frac{a}{b}\)>1 thì \(\frac{a}{b}\)<\(\frac{a-n}{b-n}\)(nEN)

CM công thức:

Ta có \(\frac{a}{b}\)>1=>a>b=>a=b+n(nEN)

Ta so sánh \(\frac{a}{b}\) và \(\frac{a-n}{b-n}\)(nEN)

Mà a*(b-n)=ab-an=ab-(b+n)*n=ab-(bn+n²)=ab-bn-n²

       b*(a-n)=ba-bn

Vì ab-bn-n²<ba-bn

=>\(\frac{a}{b}\)<\(\frac{a-n}{b-n}\)

\(P=\dfrac{3^{2016}-6^{2016}+9^{2016}-12^{2016}+15^{2016}-18^{2016}}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\dfrac{\left(3^{2016}-6^{2016}\right)+\left(9^{2016}-12^{2016}\right)+\left(15^{2016}-18^{2016}\right)}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\dfrac{3^{2016}\left(1-2^{2016}\right)+3^{2016}\left(3^{2016}-4^{2016}\right)+3^{2016}\left(5^{2016}-6^{2016}\right)}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\dfrac{3^{2016}\left(1-2^{2016}+3^{2016}-4^{2016}+5^{2016}-6^{2016}\right)}{-\left(1^{2016}-2^{2016}+3^{2016}-4^{2016}+5^{2016}-6^{2016}\right)}\)

\(=-3^{2016}\).

Vậy \(P=-3^{2016}\)

\(2^{2016}+4^{2016}+6^{2016}+...+20^{2016}=2^{2016}\left(1+2^{2016}+3^{2016}+...+10^{2016}\right)\)

Do đó:

\(A=\frac{1^{2016}+2^{2016}+3^{2016}+...+10^{2016}}{2^{2016}+4^{2016}+6^{2016}+...+20^{2016}}=\frac{1}{2^{2016}}\)

#)Giải :

   3 x 2016 + 9 x 2016 + 20 x 2016 + 28 x 2016 + 38 x 2016 + 2016 + 2016

= ( 3 + 9 + 20 + 28 + 38 + 1 + 1 ) x 2016

= 100 x 2016

= 201600

3*2016+9*2016+20*2016+28*2016+38*2016+2016+2016

= 2016 x (3 + 9 + 20 + 28 + 38 + 1 +1)

= 2016 x (12 + 20 + 28 + 38 + 2)

= 2016 x (12 + 28 + 38 + 2 + 20)

= 2016 x ( 40 + 40 + 20)

= 2016 x 100

= 201600

\(P=\frac{3^{2016}-6^{2016}+9^{2016}-12^{2016}+15^{2016}-18^{2016}}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\frac{\left(1.3\right)^{2016}-\left(2.3\right)^{2016}+\left(3.3\right)^{2016}-\left(4.3\right)^{2016}+\left(5.3\right)^{2016}-\left(6.3\right)^{2016}}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\frac{1^{2016}.3^{2016}-2^{2016}.3^{2016}+3^{2016}.3^{2016}-4^{2016}.3^{2016}+5^{2016}.3^{2016}-6^{2016}.3^{2016}}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=\frac{-3^{2016}\left(-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}\right)}{-1^{2016}+2^{2016}-3^{2016}+4^{2016}-5^{2016}+6^{2016}}\)

\(=-3^{2016}\)

2016 × 35 + 2016 × 65

= 2016 × (35+65)

= 2016 × 100

= 201600

2016 × 99 + 2016

= 2016 × 99 + 2016 × 1 

= 2016 × (1+99)

= 2016 × 100

= 201600

2016 × 102 - 2016 × 2

= 2016 × (102-2)

= 2016 × 100

= 201600

2016 × 101 - 2016

= 2016 × 101 - 2016 × 1

= 2016 × (101-1)

= 2016 × 100

= 201600

2016 x 35 + 2016 x 65 

= 2016 x ( 35 + 65 )

= 2016 x 100

= 201600

2016 x 99 + 2016

= 2016 x 99 + 2016 x 1

= 2016 x ( 99 + 1 )

= 2016 x 100

= 201600

2016 x 102 - 2016 x 2 

= 2016 x  ( 102 - 2 )

= 2016 x 100

= 201600

2016 x 101 - 2016

= 2016 x 101 - 2016 x 1

= 2016 x ( 101 - 1 )

= 2016 x 100

= 201600

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

\(B=\frac{2016^{2016}}{2016^{2016}-3}=\frac{2016^{2016}-3+3}{2016^{2016}-3}=\frac{2016^{2016}-3}{2016^{2016}-3}+\frac{3}{2016^{2016}-3}=1+\frac{3}{2016^{2016}-3}\)

ta thấy:2016²⁰¹⁶-1>2016²⁰¹⁶-3

=>\(\frac{3}{2016^{2016}-1}<\frac{3}{2016^{2016}-3}\)

=>\(1+\frac{3}{2016^{2016}-1}<1+\frac{3}{2016^{2016}-3}\)

=>A<B
 

A = 2016²⁰¹⁶ + 2/2016²⁰¹⁶-1 = 2016²⁰¹⁶ - 1 + 3/2016²⁰¹⁶ - 1

                                           = 2016²⁰¹⁶ - 1/2016²⁰¹⁶ - 1 + 3/2016²⁰¹⁶ - 1

                                           = 1 + 3/2016²⁰¹⁶ - 1       (không biết ghi hỗn số)

B = 2016²⁰¹⁶/2016²⁰¹⁶ -3 = 2016²⁰¹⁶ - 3 + 3/2016²⁰¹⁶ - 3

                                      = 2016²⁰¹⁶ - 3/2016²⁰¹⁶ - 3 + 3/2016²⁰¹⁶ - 3

                                      = 1 + 3/2016²⁰¹⁶ - 3

So sánh : 1 + 3/2016²⁰¹⁶ - 1 và 1 + 3/2016²⁰¹⁶ - 3

Ta có : 1 + 3/2016²⁰¹⁶ - 1 < 1 + 3/2016²⁰¹⁶ - 3

=> A < B