how to prove $1/n (1-(1/2)^n)$ decreasing without using differentiation

Question

$a(n)=1/n (1-(1/2)^n)$ prove $a(n+1)<a(n)$ for n>0 by differentiating slope comes negative and then we can prove it . but i wanted to solve it without that . can someone help

Answer 1

\begin{align} a(n+1)&=\frac{1}{(n+1)(1-(1/2)^{n+1})} \\ &<\frac{1}{n(1-(1/2)^{n+1})} \\ &=\frac{1}{n(1-(1/2)^{n})+n(1/2)^n/2} \\ &<\frac{1}{n(1-(1/2)^n)}=a(n) \end{align}

Answer 2

$$ a(n) - a(n+1) = \frac{1}{n} \biggl( \biggl(1 - \frac{1}{2^n} \biggr) - \biggl( 1 - \frac{1}{n+1} \biggr) \biggl( 1 - \frac{1}{2^{n+1}} \biggr) \biggr) = \frac{1}{n} \biggl( \frac{1}{n+1}\biggl( 1 - \frac{1}{2^{n+1}} \biggr) - \frac{1}{2^{n+1} } \biggr) = \frac{2^{n+1} - (n+2)}{2^{n+1}n(n+1)} > 0 $$

Answer 3

$$a\left(n+1\right)<a\left(n\right)\iff n\left(2^{n+1}-1\right)<\left(n+1\right)\left(2^{n+1}-2\right)\iff n+2<2^{n+1}$$