Question

Pentru punctele b și c va rog

Answer 1

$b)2^5*x^5=(\sqrt[5]{2^5} +\sqrt[5]{x^5})^5=(2+x)^5\\ 2^5*x^5*(243x^5)=(2+x)^5*3^5x^5=(\sqrt[5]{(2+x)^5} +\sqrt[5]{3^5x^5})^5=(2+x+3x)^5=(2+4x)^5\\ 2^5*x^5*(243x^5)=100000<=>(2+4x)^5=10^5=>2+4x=10<=>4x=8|:4=>x=2$

$c)M=1^5*2^5*...*10^5\\1^5*2^5=(\sqrt[5]{1^5} +\sqrt[5]{2^5})^5=(1+2)^5\\ P(n):1^5*2^5*...*n^5=(1+2+...+n)^5,n\geq 2\\1.Verificarea:n=2=>1^5*2^5=(1+2)^5<=>(\sqrt[5]{1^5} +\sqrt[5]{2^5})^5=(1+2)^5<=>(1+2)^5=(1+2)^5(A)\\2.Demonstratia:P(k)->P(k+1),k\geq 2\\v(P(k))=1,k\geq 2\\P(k):1^5*2^5*...*k^5=(1+2+..+k)^5,k\geq 2\\P(k+1):1^5*2^5*...*k^5*(k+1)^5=(1+2+..+k+k+1)^5,k\geq 2$

$1^5*2^5*...*(k+1)^5=(1^5*2^5*...*k^5)*(k+1)^5=(1+2+...+k)^5*(k+1)^5=(\sqrt[5]{(1+2+...+k)^5}+\sqrt[5]{(k+1)^5})^5=(1+2+...+k+k+1)^5(A),k\geq 2=>v(P(k+1))=1,k\geq 2=>P(k)->P(k+1),k\geq 2=>v(P(n))=1,n\geq 2$

$1^5*2^5*...*n^5=(1+2+...+n)^5\\Pt.n=10=>1^5*2^5*...*10^5=(1+2+...+10)^5=(\frac{10(10+1)}{2})^5=(\frac{10*11}{2}) ^5=(5*11)^5=5^5*11^5\\M=1^5*2^5*...*10^5=5^5*11^5\\N=5^5*11^5\\M-N=1^5*2^5*...*10^5-5^5*11^5=5^5*11^5-5^5*11^5=0=>M-N=0$