Question

If ^n +5P_n +1 =\[\frac{11 (n - 1)}{2}\]^n +3P_n, find n.

Answer 1

^n +5P_n +1 =_11(n-1)2^n +3P_n

\[\Rightarrow \frac{\left( n + 5 \right)!}{\left( n + 5 - n - 1 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{\left( n + 3 \right)!}{\left( n + 3 - n \right)!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)!}{4!} = \frac{11\left( n - 1 \right)}{2} \times \frac{\left( n + 3 \right)!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4!}{3!}\]
\[ \Rightarrow \frac{\left( n + 5 \right)\left( n + 4 \right)\left( n + 3 \right)!}{\left( n + 3 \right)!} = \frac{11\left( n - 1 \right)}{2} \times \frac{4 \times 3!}{3!}\]
\[ \Rightarrow \left( n + 5 \right)\left( n + 4 \right) = 22\left( n - 1 \right)\]
\[ \Rightarrow n^2 + 9n + 20 = 22n - 22\]
\[ \Rightarrow n^2 - 13n + 42 = 0\]
\[ \Rightarrow n = 7, 6\]