If n is a positive integer and if Δ=| n ! (n+1) ! (n+2) ! (n+1) ! (n+2

Name: Problem 7, Chapter 2: IIT JEE Super Course in Mathematics ALGEBRA II
Uploaded: 2021-06-13T17:09:24-08:00
Duration: 9 min 48 s
Channel: Anas Venkitta
Description: If n is a positive integer and if Δ=| n ! (n+1) ! (n+2) ! (n+1) ! (n+2) ! (n+3) ! (n+2) ! (n+3) ! (n+4) ! |, then show that {(Δ)/((n !)^3)-4} is divisible by n .

If n is a positive integer and if Δ=| n ! (n+1)...

$$ \text { If } \mathrm{n} \text { is a positive integer and if } \Delta=\left|\begin{array}{ccc} \mathrm{n} ! & (\mathrm{n}+1) ! & (\mathrm{n}+2) ! \\ (\mathrm{n}+1) ! & (\mathrm{n}+2) ! & (\mathrm{n}+3) ! \\ (\mathrm{n}+2) ! & (\mathrm{n}+3) ! & (\mathrm{n}+4) ! \end{array}\right|, \text { then show that }\left\{\frac{\Delta}{(\mathrm{n} !)^{3}}-4\right\} \text { is divisible by } \mathrm{n} \text { . } $$

$$
\text { If } \mathrm{n} \text { is a positive integer and if } \Delta=\left|\begin{array}{ccc}
\mathrm{n} ! & (\mathrm{n}+1) ! & (\mathrm{n}+2) ! \\
(\mathrm{n}+1) ! & (\mathrm{n}+2) ! & (\mathrm{n}+3) ! \\
(\mathrm{n}+2) ! & (\mathrm{n}+3) ! & (\mathrm{n}+4) !
\end{array}\right|, \text { then show that }\left\{\frac{\Delta}{(\mathrm{n} !)^{3}}-4\right\} \text { is divisible by } \mathrm{n} \text { . }
$$

Key Concepts

Determinants and Matrix Operations

This concept involves understanding the properties and computation methods of determinants, such as the effects of row and column operations, expansion by minors, and factoring common terms from rows or columns. Mastery of these operations is crucial when dealing with matrices whose elements follow specific patterns or involve complex expressions.

Factorial Function Properties

The factorial function plays an important role in combinatorics and sequences, defined recursively and exhibiting rapid growth. Recognizing patterns in factorial expressions and knowing how to simplify expressions that involve factorials are essential skills in algebraic manipulation and simplification.

Divisibility in Number Theory

Divisibility is a fundamental concept in number theory that deals with determining whether one integer is a factor of another. Techniques in divisibility include using modular arithmetic, factorization, and algebraic manipulation to prove that a given expression is divisible by a particular integer.

Algebraic Simplification and Transformation

Algebraic manipulation involves rewriting and simplifying complex expressions to reveal underlying structures, such as common factors and patterns. This process often makes it easier to apply number theoretic or combinatorial arguments to conclude properties like divisibility or integrality.

Transcript

00:01 So in the given question we are told that if n is a positive integer, if n is a positive indiger then we are told that we are given another variable that is delta which is given as the determinant of the determinant of n factorial, n plus 1 factorial, n plus 2 factorial, and in the second row we have n plus 1 factorial, n plus 2 factorial, n plus 3 factorial.

01:00 In the third row we have n plus 2 factorial n plus 3 factorial and n plus 4 factorial so this is what we have and we are told to show that we are told to show that delta divided by n factorial raised to the power 3 minus 4 is divisible by n this is what we need to show so let's take the determinant that is given in the question and we can write we can simplify this determinant further by writing it as n factorial n plus 1 times n factorial n plus 2 times n plus n factorial in the second row we have n plus 1 factorial n plus 2 times n plus 1 factorial n plus 3 times n plus 2 times n plus 1 factorial this is the expanded form of the second row and in the last row we have n plus 2 factorial times x n plus 2 factorial the next element is n plus 3 times n plus 2 factorial and then we have n plus 4 times n plus 3 times n plus 2 factorial right so now what we can do over here is to take n factorial as a common factor from the first row of this determinant we can take n plus 1 factorial from the 2nd row and we can take n plus 2 factorial from the third row and then what we would have is 1 n plus 1 n plus 2 times n plus 1 in the first row second row you would have 1 n plus 2 n plus 3 times n plus 2 last row 1 and plus 3 and plus 3 n plus 4 times n plus 3.

03:43 So this is what we have as the simplified form, right? and now what we are doing is we are going to perform some row operations on this determinant and the first operation is r2 changes to r2 minus r1 and then we are going to do the operation r3 changes to r3 minus r1...

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