The sides of a triangle are $a-1,a,a+1$  where $a$  is a natural number and $a>1$, its largest angle is twice the smallest. The value of  $3a^3+20a^2+35a+1$ is equal to

Let $\alpha$ be the smallest angle then $2\alpha$ is largest. So from the triangle we have

$\frac{\sin \alpha }{a-1}=\frac{\sin 2\alpha }{a+1}\Rightarrow \frac{a+1}{a-1}=2\cos \alpha$

Also $\cos \alpha =\frac{{\left(a+1\right)}^2+a^2-{\left(a-1\right)}^2}{2\left(a+1\right)a}=\frac{a^2+4a}{2\left(a+1\right)a}$

$\therefore \frac{a+1}{a-1}=\frac{a+4}{a+1}\Rightarrow a=5$

Hence $3a^3+20a^2+35a+1=375+500+175+1=1051$