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Given-u-0-5-u-n-1-3u-n-4-1-show-that-n-N-u-n-2-3-u-n-1-a-Deduct-that-u-n-is-odd-c-Show-that-GCD-u-n-u-n-1-1-d-Deduct-GCD-6-3-1002-6-3-1003-GCD-greate




Question Number 124682 by mathocean1 last updated on 05/Dec/20
Given  { ((u_0 =5)),(( u_(n+1) =3u_n āˆ’4)) :}  1. show that āˆ€ nāˆˆN, u_n =2+3^u_(n+1)       a. Deduct that u_n  is odd.  c. Show that GCD(u_n ;u_(n+1) )=1  d. Deduct GCD(6+3^(1002) ;6+3^(1003) ).    GCD=greatest common divisor^
$${Given}\:\begin{cases}{{u}_{\mathrm{0}} =\mathrm{5}}\\{\:{u}_{{n}+\mathrm{1}} =\mathrm{3}{u}_{{n}} āˆ’\mathrm{4}}\end{cases} \\ $$$$\mathrm{1}.\:{show}\:{that}\:\forall\:{n}\in\mathbb{N},\:{u}_{{n}} =\mathrm{2}+\mathrm{3}^{{u}_{{n}+\mathrm{1}} } \:\:\: \\ $$$${a}.\:{Deduct}\:{that}\:{u}_{{n}} \:{is}\:{odd}. \\ $$$${c}.\:{Show}\:{that}\:{GCD}\left({u}_{{n}} ;{u}_{{n}+\mathrm{1}} \right)=\mathrm{1} \\ $$$${d}.\:{Deduct}\:{GCD}\left(\mathrm{6}+\mathrm{3}^{\mathrm{1002}} ;\mathrm{6}+\mathrm{3}^{\mathrm{1003}} \right). \\ $$$$ \\ $$$${GCD}={greatest}\:{common}\:{divisor}^{} \\ $$

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