a) [latex]a_{n+1} - a_n = 5(n + 1)^{2}-12(n+1)+8 - (5n^{2}-12n+8) = 5[(n + 1)^2 - n^2] - 12[(n + 1) - n] = 5(2n + 1) - 12 = 10n - 7\ n geq 1 Rightarrow a_{n+1} - a_n > 0[/latex] ciąg ściśle rośnie b) [latex]a_{n + 1} - a_n= frac{2^{n+1}}{2(n+1)+1} - frac{2^{n}}{2n+1} = 2^nleft(frac{2}{2n + 3} - frac{1}{2n + 1} ight) = 2^nleft(frac{2(2n + 1) - (2n + 3)}{(2n + 3)(2n + 1)} ight) = 2^nleft(frac{2n - 1}{(2n + 3)(2n + 1)} ight) geq 2^nleft(frac{1}{(2n + 3)(2n + 1)} ight) > 0[/latex] ciąg ściśle rośnie c) [latex]a_{n+1} - a_n = sqrt{(n+1) + 1} - sqrt{n + 1} - (sqrt{n+1} - sqrt{n}) = sqrt{n + 2} + sqrt{n} - 2sqrt{n+1} =[/latex] [latex]= (sqrt{n + 2} + sqrt{n} - 2sqrt{n+1})*1 = (sqrt{n + 2} + sqrt{n} - 2sqrt{n+1}) * frac{sqrt{n + 2} + sqrt{n} + 2sqrt{n+1}}{sqrt{n + 2} + sqrt{n} + 2sqrt{n+1}} = [/latex] [latex]frac{(sqrt{n + 2} + sqrt{n})^2 - (2sqrt{n+1})^2}{sqrt{n + 2} + sqrt{n} + 2sqrt{n+1}} = frac{(2n + 2 + 2sqrt{n(n + 2)}) - (4n + 4)}{ sqrt{n + 2} + sqrt{n} + 2sqrt{n+1}} = frac{2sqrt{n(n + 2)} - (2n + 2)}{ sqrt{n + 2} + sqrt{n} + 2sqrt{n+1}} = [/latex] [latex]= frac{(2sqrt{n(n + 2)})^2 - (2n + 2)^2}{(sqrt{n + 2} + sqrt{n} + 2sqrt{n+1})(2sqrt{n(n + 2)} + (2n + 2))}= frac{(4n^2 + 8n) - (4n^2 + 8n + 4)}{(sqrt{n + 2} + sqrt{n} + 2sqrt{n+1})(sqrt{n(n + 2)} + (2n + 2))} = [/latex] [latex]frac{- 4}{(sqrt{n + 2} + sqrt{n} + 2sqrt{n+1})(sqrt{n(n + 2)} + (2n + 2))}< 0[/latex] ciąg ściśle maleje
