Question

1)Arătați ca :2^1×2^2×2^3×.....×2^100+(16^2)^631=5×4^2524
2)Sa se scrie numărul 101^n,n €Nu ca o suma de 101 numere naturale consecutive

Answer 1

$2^{1}*2^{2}*2^{3}*...*2^{100}+(16^{2})^{631}=$

$=2^{1+2+3+...+100}+(2^{4})^{2*631}=$

$=2^{100*101:2}+(2^{4})^{1262}=$

$=2^{5050}+2^{5048}=$

$=2^{5048}(2^{2}+1)=$

$=(2^{2})^{2524}*5=$

$=5*4^{2524}$




$101^{n}=$

$=101^{n-1}*101=$

$=101^{n-1}(1+1+1+....+1)=$
(in paranteza 1 se repeta de 101 ori)

$=101^{n-1}+101^{n-1}+...+101^{n-1}$
($101^{n-1}$ se repeta de 101 ori)