Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 211437 by necx122 last updated on 09/Sep/24

Commented by AlagaIbile last updated on 09/Sep/24

$$Ipostedthequestionon$$$$CowbellpediaWhatshappgrouptwo$$$$hoursago.Howdiditreachhere?$$

Answered by A5T last updated on 09/Sep/24

$$x^2=x+1\Rightarrow x^4=x^2+2x+1=(x+1)+2x+1=3x+2$$$$\Rightarrow x^8=9x^2+12x+4=9(x+1)+12x+4=21x+13$$$$\Rightarrow x^{16}=441\left(x^2\right)+546x+169=987x+610$$$$\Rightarrow x^{20}=x^{16}\times x^4=(987x+610)(3x+2)$$$$=2961x^2+3804x+1220=6765x+4181=ax+b$$$$\Rightarrow a+b=10946$$

Answered by Rasheed.Sindhi last updated on 09/Sep/24

$$x^2=x+1\Rightarrow x^4=x^2+2x+1$$$$=(x+1)+2x+1=3x+2$$$$\Rightarrow x^5=3x^2+2x=3(x+1)+2x=5x+3$$$$$$$$x^{20}={\left(x^5\right)}^4={\left\{{(5x+3)}^2\right\}}^2$$$$={(25x^2+30x+9)}^2$$$$={\{25(x+1)+30x+9\}}^2$$$$={(55x+34)}^2$$$$=3025x^2+3740x+1156$$$$=3025(x+1)+3740x+1156$$$$=6765x+4181$$$$a=6765,b=4181$$$$a+b=10946$$

Commented by necx122 last updated on 09/Sep/24

$$Thididgreat$$

Answered by A5T last updated on 09/Sep/24

$$Observethat{x}^2-x-1=0isthecharacteristic$$$$equationof{f}_n=f_{n-1}+f_{n-2}$$$$\Rightarrow x^n=f_{n}x+f_{n-1}\Rightarrow x^{20}=f_{20}x+f_{19}$$$$\Rightarrow a+b=f_{20}+f_{19}=f_{21}=10946$$$$f_0=0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,$$$$987,1597,2584,4181,6765,f_{21}=10946$$

Commented by necx122 last updated on 09/Sep/24

$$hmmm..neverknewthis.Coolstuff.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com