Question

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

Answer 1

Let $\alpha$ be the smallest angle then 2$\alpha$, is largest.
So from the sine law. we have
$\frac{sin\alpha }{a-1}=\frac{sin2\alpha }{a+1}$
$\Rightarrow \frac{a+1}{a-1}=2cos\alpha$
Also $cos\alpha =\frac{(a+1{)}^2+a^2-(a-1{)}^2}{2(a+1)a}=\frac{a^2+4a}{2(a+1)a}$
$\therefore \frac{a+1}{a-1}=\frac{a+4}{a+1}$
$\Rightarrow a=5$
Hence $3a^2+30a^2+35a+1=375+500+175+1=1051$