Question

cum faci 2 la puterea a 15 minus 2 la puterea 14 minus ... minus 2 la puterea 0

Answer 1

 

$\displaystyle\bf\\Fosim~formula:~~~2^0+2^1+2^2+...+2^n=2^{n+1}-1\\\\Formula~generala~este:~~k^0+k^1+k^2+...+k^n=\frac{k^{n+1}-1}{k-2}\\\\Rezolvare:\\\\2^{15}-2^{14}-...2^0=\\\\=2^{15}-2^{14}-2^{13}-2^{12}-...-2^0=\\\\=2^{15}+\Big(-2^{14}-2^{13}-2^{12}-...-2^0\Big)=\\\\=2^{15}-\Big(2^{14}+2^{13}+2^{12}+...+2^0\Big)=\\\\=2^{15}-\Big(2^0+2^1+2^2+...2^{14}\Big)=\\\\=2^{15}-\Big(2^{14+1}-1\Big)=\\\\=2^{15}-\Big(2^{15}-1\Big)=\\\\=2^{15}-2^{15}+1=\boxed{\bf1}$