ma10.mw

> (1+(n+1)*a)^n < (1+n*a)^(n+1)
 

(1+(n+1)*a)^n < (1+n*a)^(n+1) 

> L := proc (n) options operator, arrow; (1+n*a)^(n+1) end proc; 1
 

proc (n) options operator, arrow; (1+n*a)^(n+1) end proc 

> R := proc (n) options operator, arrow; (1+(n+1)*a)^n end proc; 1
 

proc (n) options operator, arrow; (1+(n+1)*a)^n end proc 

> Ldiv = L(k)/L(k-1); 1
 

(1+k*a)*(1+a/(1+(k-1)*a))^k = (1+k*a)^(k+1)/(1+(k-1)*a)^k 

> Ldiv = (1+k*a)*((1+(k-1)*a+a)/(1+(k-1)*a))^k
 

(1+k*a)*(1+a/(1+(k-1)*a))^k = (1+k*a)*((1+(k-1)*a+a)/(1+(k-1)*a))^k 

> Ldiv := (1+k*a)*(1+a/(1+(k-1)*a))^k
 

(1+k*a)*(1+a/(1+(k-1)*a))^k 

> Rdiv = R(k)/R(k-1); 1
 

(1+k*a)*(1+a/(1+k*a))^k = (1+(k+1)*a)^k/(1+k*a)^(k-1) 

> Rdiv := (1+k*a)*(1+a/(1+k*a))^k
 

(1+k*a)*(1+a/(1+k*a))^k 

> Rdiv < Ldiv
 

(1+k*a)*(1+a/(1+k*a))^k < (1+k*a)*(1+a/(1+(k-1)*a))^k 

> 1/(1+k*a) < 1/(1+(k-1)*a)
 

1/(1+k*a) < 1/(1+(k-1)*a) 

> A to samy desetkrat rychleji:
(derivace podle a ze zlogritmovanych tvaru stran)
 

> diff(ln((1+(n+1)*a)^n), a) < diff(ln((1+n*a)^(n+1)), a)
 

n*(n+1)/(1+(n+1)*a) < (n+1)*n/(1+n*a) 

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