Question

1+15+15*16+15×16^2+....+15×6^m=1+3+3×4+3×4^2+.....+3×4^99 repede dau coroana

Answer 1

Răspuns:

Explicație pas cu pas:

$L.H.S.~ 1+15+15\cdot 16+15\cdot 16^2+\ldots+15\cdot 16^m = \\= 1+(16-1)+(16-1)\cdot 16+(16-1)\cdot 16^2+\ldots+(16-1)\cdot 16^m=\\1+16-1+16^2-16+16^3-16^2+\ldots+16^{m+1}-16^m=\\\texttt{Termenii se reduc si in final mai ramane }16^{m+1}\\\texttt{Analog, pentru R.H.S. obtinem }4^{100}\\\texttt{Ecuatia devine }16^{m+1}=4^{100}\\4^{2m+2}=4^{100}\\2m+2=100\\2m=98\\\boxed{m=49}$