Question

Giúp mình nhé

Mình cảm ơn

Cho B= 3 + 3³ + 3⁵ +....+ 3^1991. Chứng minh rằng B chia hết cho 13 và 41

 

 

Answer 1

Lời giải:

\(B=3+3^3+3^5+...+3^{1991}=(3+3^3+3^5)+(3^7+3^9+3^{11})+....+(3^{1987}+3^{1989}+3^{1991})\)

\(=3(1+3^2+3^4)+3^7(1+3^2+3^4)+....+3^{1987}(1+3^2+3^4)\)

\(=(1+3^2+3^4)(3+3^7+3^{13}+...+3^{1987})\)

\(=91(3+3^7+3^{13}+...+3^{1987})\)

\(=13.7(3+3^7+3^{13}+.....+3^{1987})\vdots 13\) (đpcm)

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\(B=(3+3^3+3^5+3^7)+....+(3^{1985}+3^{1987}+3^{1989}+3^{1991})\)

\(=3(1+3^2+3^4+3^6)+....+3^{1985}(1+3^2+3^4+3^6)\)

\(=(1+3^2+3^4+3^6)(3+...+3^{1985})=820(3+...+3^{1985})=41.20(3+...+3^{1985})\vdots 41\) (đpcm)