Cho B=3+3^2+3^3+.....+3^2015+3^2016. Chứng minh rằng : a.B:4
b.B:13
c.B:40
K
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KT
25 tháng 7 2018
\(A=1+3+3^2+3^3+3^4+...+3^{2015}\)
\(=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+...+\left(3^{2012}+3^{2013}+3^{2014}+3^{2015}\right)\)
\(=\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)+...+3^{2012}\left(1+3+3^2+3^3\right)\)
\(=\left(1+3+3^2+3^3\right)\left(1+3^4+...+3^{2012}\right)\)
\(=40\left(1+3^4+...+3^{2012}\right)\)\(⋮\)\(5\)
\(B=2+2^2+2^3+...+2^{2016}\)
\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{2013}+2^{2014}+2^{2015}+2^{2016}\right)\)
\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+..+2^{2013}\left(1+2+2^2+2^3\right)\)
\(=\left(1+2+2^2+2^3\right)\left(2+2^5+...+2^{2013}\right)\)
\(=15\left(2+2^5+...+2^{2013}\right)\)\(⋮\)\(15\)
LD
1
21 tháng 12 2017
A = 31 + 32 +33 + 34 +.....+32015+ 32016
A = (31 + 32) +(33 + 34) +.....+ (32015+ 32016)
A = 3(1+3) + 32(1+3) + .....+ 32015(1+3)
A = 3.4 +32.4 +....... + 32015.4
A = 4(3 +32 +....+ 32015) chia hết cho 4
===================================================
A =31 + 32 +33 + 34 + 35 +36 +.....+32014 + 32015+ 32016
A = (31 + 32 +33 ) +(34 + 35 +36) +.....+ (32014 + 32015+ 32016)
A = 3(1+3+32) + 34(1+3+32) + .....+ 32014(1+3+32)
A = 3.13 +34.13 +....... + 32014.13
A = 13.(3 +34 +....+ 32014) chia hết cho 13
MK
1
TN
1
10 tháng 1 2016
a,3B=3+32+33+............+32016
b,3B-B=32016-1
=>2B=32016-1
=>B=32016-1:2
=>đpcm
14 tháng 3 2022
Ta có : \(\dfrac{1}{2^2}\)<\(\dfrac{1}{1.2}\); \(\dfrac{1}{3^2}\)<\(\dfrac{1}{2.3}\);.....;\(\dfrac{1}{2016^2}\)<\(\dfrac{1}{2015.2016}\)
⇒ A = \(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}\)+...+\(\dfrac{1}{2016^2}\)< \(\dfrac{1}{1.2}\)+\(\dfrac{1}{2.3}\)+...+\(\dfrac{1}{2015.2016}\)
⇒ A = \(\dfrac{1}{2^2}\)+\(\dfrac{1}{3^2}\)+...+\(\dfrac{1}{2016^2}\) < 1 - \(\dfrac{1}{2016}\)= \(\dfrac{2015}{2016}\) (ĐCPCM)