Question

Use mathematical induction to prove
$1+2+3+\dots +n<\frac{1}{8}{\left(2n+1\right)}^2$

Answer 1

$1+2+3+\dots +n<\frac{1}{8}{\left(2n+1\right)}^2$
Let $n=1$
$\begin{array}{c}P\left(1\right):1<\frac{1}{8}{\left(2\left(1\right)+1\right)}^2\\ 1<\frac{1}{8}\left(9\right)\end{array}$
is true
$\begin{array}{c}n=k\\ P\left(k\right):1+2+3+\cdots +k<\frac{1}{8}{\left(2k+1\right)}^2\end{array}$
when $n=k+1$
$\begin{array}{c}1+2+3+\cdots +k+\left(k+1\right)<\frac{1}{8}{\left(2k+1\right)}^2+\left(k+1\right)\\ <\frac{1}{8}\left[4k^2+12k+9\right]\\ <\frac{1}{8}{\left[2k+3\right]}^2\\ <\frac{1}{8}{\left[2\left(k+1\right)+1\right]}^2\end{array}$
Hence proved