Sequences & Series
GP as Triangle Sides — Floor Function
nta_pyq_2024_jan
Grade 11
Question:
If three successive terms of a G.P. with common ratio $r(r>1)$ are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r]+[-r]$ is equal to:
Step-by-Step Solution
Key Concept: Sides $a,ar,ar^2$. Triangle inequalities with $r>1$: most restrictive is $a+ar>ar^2\Rightarrow r^2-r-1<0\Rightarrow r<\frac{1+\sqrt5}{2}$. Also $r>1$. So $r\in\left(1,\frac{1+\sqrt5}{2}\right)$.
$[r]=1$, $[-r]=-2$. $3(1)+(-2)=1$.
Correct Answer: 1