Question

Prove the following by mathematical induction $1+2+3+\cdots +n<\frac{1}{8}{\left(2n+1\right)}^2$

Answer 1

$1+2+3+\cdots +n<\frac{1}{8}{\left(2n+1\right)}^2$

Let, $n=1$

$p\left(1\right):1<\frac{1}{8}{\left(2\left(1\right)+1\right)}^2$

$1<\frac{1}{8}\left(9\right)$

$n=k$

$p\left(k\right):1+2+3+........+k<\frac{1}{8}(2k+1{)}^2$

when $n=k+1$

$\begin{array}{c}1+2+3+\cdots +k+(k+1)\\ <\frac{1}{8}(2k+1{)}^2+(k+1)\\ <\frac{1}{8}\left[4k^2+1+4k+8k+8\right]\\ <\frac{1}{8}(4k^2+12k+9)\\ <\frac{1}{8}(2k+3{)}^2\\ <\frac{1}{8}\left[2\right(k+1)+1{]}^2\end{array}$