Factorials
5. Some Illustration Examples :
Examples: (i) Find n, if \(
\left( {n + 1} \right)! = 12 \times \left( {n - 1} \right)n
\)
Solution:\(
\left( {n + 1} \right)! = 12 \times \left( {n - 1} \right)!
\)
\(
\Rightarrow \left( {n + 1} \right)n! = 12 \times \left( {n - 1} \right)!
\)
\(
\Rightarrow \left( {n + 1} \right) \times n \times \left( {n - 1} \right)! = 12 \times \left( {n - 1} \right)!
\)
\(
\Rightarrow n\left( {n + 1} \right) = 12
\)
\(
\Rightarrow n^2 + n - 12 = 0
\)
\(
\Rightarrow \left( {n + 4} \right)\left( {n - 3} \right) = 0
\)
\(
\Rightarrow n = - 4
\) or 3
\(
\Rightarrow n = 3
\) since n can not be negative
Example (ii)
Prove that \(
\left( {n!} \right)^2 < n^n .n! < \left( {2n} \right)!
\)
For all positive integers n.
Solution: \(
\left( {n!} \right)^2 = \left( {n!} \right)\left( {n!} \right)
\)
\(
= \left( {1 \times 2 \times 3 \times ... \times \left( {n - 1} \right) \times n} \right)\left( {n!} \right)
\)
Now, \(
1 \leqslant n,2 \leqslant n,3 \leqslant n,....n \leqslant n
\)
\(
\Rightarrow 1 \times 2 \times 3 \times ....\left( {n - 1} \right)n \leqslant n \times n \times n \times ... \times n
\)
\(
\Rightarrow n! \leqslant n^n
\)
\(
\Rightarrow \left( {n!} \right)\left( {n!} \right) \leqslant \left( {n!} \right)n^n
\)
\(
\Rightarrow \left( {n!} \right)^2 \leqslant n^n \left( {n!} \right)
\)…………...(1)
Also,
\(
\left( {2n} \right)! = 1 \times 2 \times 3 \times ..............n \times \left( {n + 1} \right) \times ....\left( {2n - 1} \right) \times \left( {2n} \right)
\)
Now,
\(
n + 1 > n,n + 2 > n,n + 3 > n..........n + n > n
\)
\(
\Rightarrow \left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right).............\left( {2n - 1} \right)\left( {2n} \right) > n^n
\)
\(
\Rightarrow n!\left( {n + 1} \right)\left( {n + 2} \right)..........\left( {2n - 1} \right)\left( {2n} \right) > n^n .n!
\)
\(
\Rightarrow \left( {2n} \right)! > n!.n^n
\)
\(
\Rightarrow n!.n^n < \left( {2n} \right)!
\)…………….(2)
From (1) and (2), we got
\(
\left( {n!} \right)^2 \leqslant n^n .\left( {n!} \right) < \left( {2n} \right)!
\)
Examples 3: Find thre general term of the series \(
\sum\limits_{r = 1}^n {r \times r!}
\)
Solution : Here, the general term of the series is \(
_{T_r = r \times r!}
\)
\(
= \left( {r + 1 - 1} \right)r!
\)
\(
= \left( {r + 1} \right)r! - r!
\)
\(
= \left( {r + 1} \right)! - r!
\)
\(
T_1 = 2! - 1!
\)
\(
T_2 = 3! - 2!
\)
\(
T_3 = 4! - 3!
\)…..
\(
T_n = \left( {n + 1} \right)! - n!
\)