Suppose $t_n$ is the number of triangles formed using the vertices of a regular polygon of n sides. If $t_{n+1}=t_n+28,$then $n=$

${\mathrm{t}}_{\mathrm{n}}=$ No of triangles formed using n vertices of  regular polygon

$\Rightarrow {\mathrm{t}}_{\mathrm{n}}={\mathrm{n}}_{{\mathrm{C}}_3}$

${\mathrm{t}}_{\mathrm{n}+1}={\mathrm{t}}_{\mathrm{n}}+28$

${(\mathrm{n}+1)}_{{\mathrm{C}}_3}={\mathrm{n}}_{{\mathrm{C}}_3}+28$

${(\mathrm{n}+1)}_{{\mathrm{C}}_3}-{\mathrm{n}}_{{\mathrm{C}}_3}=28$

${\mathrm{n}}_{{\mathrm{C}}_2}=28$

$\Rightarrow \frac{\mathrm{n}(\mathrm{n}-1)}{2}=28 \therefore \mathrm{n}(\mathrm{n}-1)=56=8\times 7$

$\mathrm{n}=8$