Question

Badaj monotoniczność ciągu:

a) a(n)=(5n-3)/6

b) a(n)=(3n+2)/(4n-1)

Answer 1

a) a(n)=(5n-3)/6
a n+1 =[ 5*(n+1)-3] / 6 = 5n+5-3/6 = 5n+2/6
an - an+1 =  (5n-3)/6 - 5n+2/6 =  5n-3 - 5n +2 /6 = -1/6 = const.  , ciąg malejący

b) a(n)=(3n+2)/(4n-1)
a n+1 =[3(n+1)+2] / [4(n+1)-1] = [3n+3+2]/ 4n+4-1 ] = 3n+5/4n+3
an +1 - an =  3n+5/4n+3 -  (3n+2)/(4n-1)  =  [(3n+5 )*(4n-1) - (3n+2)*(4n+3)]/(4n+3) *(4n-1) = [12n^2 -3n+20n-5 - (12n^2+9n+8n+6)] / 16n^2-4n+12n-3 = 12n^2 +17n-5 -12n^2-9n-8n-6/16n^2+8n-5 = -11/16n^2+8n-5 ≠ const.