Chứng minh rằng: S = 1/3+ 2/3^2 + ........... + 2015/3^2015 < 3/4
Những câu hỏi liên quan
Cho S=3/4^2+3/6^2+3/8^2+...+3/2014^2. Chứng minh rằng S<1006/2015
Chứng minh rằng:
1+2015+20152+20153+...+20157= 2016.(1+20152)(1+20154)
\(vt=1+2015+2015^2+2015^3+2015^4+2015^5+2015^6+2015^7\)
\(=\left(1+2015\right)+\left(2015^2+2015^3\right)+\left(2015^4+2015^5\right)+\left(2015^6+2015^7\right)\)
\(=1\left(1+2015\right)+2015^2\left(1+2015\right)+2015^4\left(1+2015\right)+2015^6\left(1+2015\right)\)
\(=\left(2015+1\right)\left(1+2015^2+2015^4+2015^6\right)\)
\(=2016\left(1+2015^2+2015^4+2015^6\right)\)
\(=2016\left[\left(1+2015^2\right)+\left(2015^4+2015^6\right)\right]\)
\(=2016\left[1\left(1+2015^2\right)+2015^{2014}\left(1+2015^2\right)\right]=vp\left(đpcm\right)\)
\(=2016\left(1+2015^{2014}\right)\left(1+2015^{2012}\right)\)
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cho S=3^1+3^2+3^3=............+3^2015
chứng minh rằng: S chia hết cho 70
cho A = 1/2^2 + 1/3^2 + 1/4^2 + ... + 1/ 2015^2 + 1/2016^2. Chứng minh rằng: A < 2015/2016
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)
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Cho S= 1+3+32+33+34+35+36+....+32015.
Chứng minh rằng: S chia hết cho 10.
\(S=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+...+\left(3^{2012}+3^{2013}+3^{2014}+3^{2015}\right)\)
\(=\left(1+3+9+27\right)+3^4.\left(1+3+3^2+3^3\right)+...+3^{2012}.\left(1+3+3^2+3^3\right)\)
\(=40+3^4.40+...+3^{2012}.40\)
\(=40.\left(1+3^4+...+3^{2012}\right)\)
\(=10.4.\left(1+3^4+...+3^{2012}\right)\text{ chia hết cho 10}\)
=> S chia hết cho 10 (đpcm).
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S=(1+3+3^2)+(3^3+3^4+3^5)+...+(3^2013+3^2014+3^2015)
S=10+3^3(1+3+6)+...+3^2013(1+3+6)
S=10+3^3.10+...+3^2013.10
S=10(3^3+...+3^2013)
Vì tích trên có thừa số 10 mà 10 chia hết cho 10 nên S chia hết cho 10
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cho A = 1/1*2+1/3*4+...+1/99*100 và B= 2015/51+2015/52+2015/53+...+2015/100. Chứng minh rằng B chia hết cho A
cho S=1+3+3^2+3^3+3^4+.....+3^2015 chứng minh S chia hết 13
SCSH: (32015- 1) : 2 = 0
Tổng: (32015+ 1) : 2 = 2
Hk tốt,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,
k nhé
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Chứng minh rằng:
S=1+3+32+33+...+32015chia hết cho 40
S = 1 + 3 + 32 + .......... + 32008 + 32009
= ( 1 + 3 ) + ( 32 + 33 ) + ............. + ( 32008 + 32009 )
= 4 + 32( 1 + 3 ) + ............ + 32008( 1 + 3 )
= 4 + 4 . 32 + .......... + 4 . 32008
= 4( 1 + 32 +......... + 32008 ) chia hết cho 4
KL:......
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Chứng minh rằng 1+2/2+3/2^2+4/2^3+....+2014/2^2013+2015/2^2014 <4