Question

If \[(n + 2)! = 2550 \times n!\] , then find the value of $n$.

Answer 1

Hint: A factorial is a function that multiplies a number by every number below it. Here, we will the factorial formula \[n! = n\left( {n - 1} \right)\left( {n - 2} \right) \ldots \ldots ..{\text{ }}\left( 3 \right)\left( 2 \right)\left( 1 \right)\] . The factorial of a positive integer is represented by the symbol \[n!\] . Next, we will substitute this formula in the given equation. Then by using the factorization method, we will find the value of n.

Complete step by step solution:
We know that,
$n! = n(n-1)(n-2)(n-3)……1$
We are given that,
\[(n + 2)! = 2550 \times n!\]
As we know that, \[(n + 2)! = (n + 2)(n + 1)n!\]
Substituting this value in the given statement, we will get,
\[ \Rightarrow (n + 2)(n + 1)n! = 2550n!\]
Here, n! is cancelled from both sides, we get,
\[ \Rightarrow (n + 2)(n + 1) = 2550\]
Removing the brackets, we get,
\[ \Rightarrow {n^2} + n + 2n + 2 = 2550\]
On evaluating this above equation, we will get,
\[ \Rightarrow {n^2} + 3n + 2 = 2550\]
By using the transposition method, moving the RHS term to LHS, we get,
\[ \Rightarrow {n^2} + 3n + 2 - 2550 = 0\]
On simplifying this above equation, we get,
\[ \Rightarrow {n^2} + 3n - 2548 = 0\]
By using the factorization method, we will find factors of this given equation as below,
\[ \Rightarrow {n^2} - 49n + 52n - 2548 = 0\]
\[ \Rightarrow n(n - 49) + 52(n - 49) = 0\]
\[ \Rightarrow (n + 52)(n - 49) = 0\]
Comparing this, we get,
\[ \Rightarrow n = - 52\] or \[n = 49\]
Since, the value for factorial n cannot be negative. Thus, we have only one value of n i.e. \[n = 49\].
Hence, if given \[(n + 2)! = 2550 \times n!\] then the value of \[n = 49\] .

Note:
Alternative Method:
Given, \[(n + 2)! = 2550 \times n!\]
\[ \Rightarrow (n + 2)(n + 1)n! = 2550n!\]
\[ \Rightarrow (n + 2)(n + 1) = 2550\]
\[ \Rightarrow (n + 2)(n + 1) = 51 \times 50\]
Comparing both the sides, we get,
\[\therefore n + 2 = 51\]
\[ \Rightarrow n = 51 - 2\]
\[ \Rightarrow n = 49\]
Next,
\[\therefore n + 1 = 50\]
\[ \Rightarrow n = 50 - 1\]
\[ \Rightarrow n = 49\]
Thus, we have the same value for n i.e. n=49.

We can find the value of n without using the factorization method. The multiplication of all positive integers, say “n”, that will be smaller than or equivalent to n is known as the factorial. The factorial operation is encountered in many areas of Mathematics such as algebra, permutation and combination, and mathematical analysis. Its primary use is to count ‘n’ possible distinct objects.