Question

prove that given sequence is monotonically decreasing​

Answer 1

there are two methods:

(1.) firstly, you may go for principle of mathematical induction which I'm not gonna diacuss here

(2.)secondly, to prove that this sequence is montonically decreasing we just have to prove that

$u \frac{}{n + 1} - u \frac{}{n} < 0 \\ \frac{3(n + 1) + 9}{2(n + 1) + 1} - (\frac{3n + 9}{2n + 1} ) < 0 \\ \frac{3n + 12}{2n + 3} - ( \frac{3n + 9}{2n + 1} ) < 0 \\ \frac{(3n + 12)(2n + 1) - (3n + 9)(2n + 3)}{(2n + 3)(2n + 1)} < 0 \\ \frac{(6 {n}^{2} + 27n + 12) - (6 {n}^{2} + 27n + 27) }{(2n + 3)(2n + 1)} < 0 \\ \frac{ - 15}{(2n + 3)(2n + 1)} < 0 \\ as \: for \: all \: n > 0 \: (2n + 3)(2n + 1) > 0 \\ therefore \\ u \frac{}{n + 1} - u \frac{}{n} < 0 \\ hence \: given \: sequence \: is \: monotonically \: decreasing$