cho A = 31+32+33 ....+32016 :chứng tỏ răng
a, A chia hết cho 4,5
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\(A=\left(3+3^2+3^3\right)+...+\left(3^{58}+3^{59}+3^{60}\right)\\ A=3\left(1+3+3^2\right)+...+3^{58}\left(1+3+3^2\right)\\ A=\left(1+3+3^2\right)\left(3+...+3^{58}\right)\\ A=13\left(3+...+3^{58}\right)⋮13\)
\(M=\left(2+2^2+2^3+2^4\right)+...+\left(2^{17}+2^{18}+2^{19}+2^{20}\right)\\ M=\left(2+2^2+2^3+2^4\right)+...+2^{16}\left(2+2^2+2^3+2^4\right)\\ M=\left(2+2^2+2^3+2^4\right)\left(1+...+2^{16}\right)\\ M=30\left(1+...+2^{16}\right)⋮5\)
Đặt A = 3¹ + 3² + 3³ + 3⁴ + ... + 3⁹⁹ + 3¹⁰⁰
= (3¹ + 3²) + (3³ + 3⁴) + ... + (3⁹⁹ + 3¹⁰⁰)
= 3.(1 + 3) + 3³.(1 + 3) + ... + 3⁹⁹.(1 + 3)
= 3.4 + 3³.4 + ... + 3⁹⁹.4
= 4.(3 + 3³ + ... + 3⁹⁹) ⋮ 4
Vậy A ⋮ 4
\(B=3^0+3^1+3^2...+3^{100}\)
\(=3^0\times\left(1+3^1+3^2\right)+3^3\times\left(1+3^1+3^2\right)+...+3^{98}\times\left(1+3^1+3^2\right)\)
\(=3^0\times13+3^3\times13+...+3^{98}\times13\)
\(=13\times\left(3^0+3^3+...+3^{98}\right)⋮13\)
Lời giải:
$A=1+(3+3^2+3^3)+(3^4+3^5+3^6)+....+(3^{2014}+3^{2015}+3^{2016})$
$=1+3(1+3+3^2)+3^4(1+3+3^2)+...+3^{2014}(1+3+3^2)$
$=1+3.13+3^4.13+....+3^{2014}.13$
$=1+13(3+3^4+...+3^{2014})$
$\Rightarrow A-1\vdots 13(1)$
Mặt khác:
$A=1+(3+3^2+3^3+3^4)+....+(3^{2013}+3^{2014}+3^{2015}+3^{2016})$
$=1+3(1+3+3^2+3^3)+....+3^{2013}(1+3+3^2+3^3)$
$=1+(3+...+3^{2013})(1+3+3^2+3^3)$
$=1+40(3+....+3^{2013})$
$\Rightarrow A-1\vdots 5(2)$
Từ $(1); (2)$ mà $(5,13)=1$ nên $A-1\vdots (5.13)$ hay $A-1\vdots 65$
$\Rightarrow A$ chia $65$ dư $1$
\(A=3+3^2+3^3+...+3^{99}\\ \Rightarrow A=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+...+\left(3^{97}+3^{98}+3^{99}\right)\\ \Rightarrow A=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+...+3^{97}\left(1+3+3^2\right)\\ \Rightarrow A=\left(1+3+3^2\right)\left(3+3^4+...+3^{97}\right)\\ \Rightarrow A=13\left(3+3^4+...+3^{97}\right)⋮13\)
\(A=3+3^2+3^3+...+3^{99}\\ 3A-A=3^{99}-1\\ A=\dfrac{3^{99}-1}{2}\)