1 problem found
Three rods have lengths \(a\), \(b\) and \(c\), where \(a< b< c\). The three rods can be made into a triangle (possibly of zero area) if \(a+b\ge c\). Let \(T_{n}\) be the number of triangles that can be made with three rods chosen from \(n\) rods of lengths \(1\), \(2\), \(3\), \(\ldots\) , \(n\) (where \(n\ge3\)). Show that \(T_8-T_7 = 2+4+6\) and evaluate \(T_8 -T_6\). Write down expressions for \(T_{2m}-T_{2m-1}\) and \(T_{2m} - T_{2m-2}\). Prove by induction that \(T_{2m}=\frac 16 m (m-1)(4m+1)\,\), and find the corresponding result for an odd number of rods.
Solution: Every \(T_7\) triangle is valid, so we are interested in new triangles which have \(8\) has a longest side. We can have: \begin{array}{c|c|c} \text{longest} & \text{middle} & \text{shortest} \\ \hline 8 & 7 & 1-6 \\ 8 & 6 & 2-5 \\ 8 & 5 & 3-4 \end{array} which is \(6+4+2\) extra triangles. The new ones excluding all the sixes are: \begin{array}{c|c|c} \text{longest} & \text{middle} & \text{shortest} \\ \hline 8 & 7 & 1-6 \\ 8 & 6 & 2-5 \\ 8 & 5 & 3-4 \\ 7 & 6 & 1-5 \\ 7 & 5 & 2-4 \\ 7 & 4 & 3 \\ \end{array} Ie \(2+4+6 + 1 + 3+5\) \(T_{2m}-T_{2m-1} = 2 \frac{(m-1)m}{2} = m(m-1)\) and \(T_{2m}-T_{2m-2} = \frac{(2m-2)(2m-1)}{2}\) \(T_4 = 3\) (\(1,2,3\), \(1,3,4\), \(2,3,4\)) and \(\frac16 \cdot 2 \cdot 1 \cdot 9 = 3\) so the base case holds. Suppose it's true for some \(m = k\), then \begin{align*} && T_{2(k+1)} &= T_{2k} + \frac{2m(2m+1)}{2} \\ &&&= \frac{m(m-1)(4m+1)}{6} + \frac{6m(2m+1)}{6}\\ &&&= \frac{m(4m^2-3m-1+12m+6)}{6} \\ &&&= \frac{m(4m^2+9m+5)}{6}\\ &&&= \frac{m(4m+5)(m+1)}{6}\\ &&&= \frac{(m+1-1)(4(m+1)+5)(m+1)}{6}\\ \end{align*} as required, therefore it is true by induction. For odd numbers, we can see that \(T_{2m-1} = \frac{m(m-1)(4m+1)}{6} - m(m-1) = \frac{m(m-1)(4m-5)}{6}\)