Question

Daca a = 1 +2 la puterea 1+2la puterea2+2la puterea3+.....+2 la puterea 59,atunci aratati ca numarul a+1 este si pătrat perfect si cub perfect

Answer 1

$a=2^{0}+2^{1}+2^{2}+2^{3}+..........+2^{n}=2^{n+1}-1 \\ a=2^{0}+2^{1}+2^{2}+2^{3}+....+2^{59}= 2^{59+1}-1 \\ a=2^{60}-1$
demonstratie ca a+1 este patrat perfect:
$a=2^{60}-1 \\ a+1=2^{60}$
$a+1=(2^{30})^{2}$
numarul este patrat perfect
demonstratie ca numarul a+1 este cub perfect:
$a=2^{60}-1 \\ a+1=2^{60} \\ a+1=(2^{20})^{3}$
numarul este cub perfect

Answer 2

$a=1+ 2^{1} + 2^{2} + 2^{3} +.....+ 2^{59} | *2 \\ \\ 2a=2+ 2^{2} + 2^{3} + 2^{4} +....+ 2^{59} + 2^{60} \\ Acum -se-vor-scadea \\ \\ 2a-a= 2^{60} -1 \\ \\ a=2^{60} -1 \\ \\ a+1=2^{60} -1 +1 \\ \\ a+1= 2^{60} \\ \\ 2^{60}= 2^{30*2}=\ \textgreater \ patrat.perfect \\ \\ 2^{60}= 2^{20*3}- cub.perfect$