Prove that
(2n)!/n!=1x3x5.........(2-1)x2n.
To prove that (2n)!/n! = 1 × 3 × 5 × ... × (2n-1) × 2n, we can use the concept of factorial and basic algebraic manipulation.
Let's start by expanding the expression (2n)! using factorial notation:
(2n)! = (2n) × (2n-1) × (2n-2) × ... × 3 × 2 × 1
Now, let's consider the expression n!:
n! = n × (n-1) × (n-2) × ... × 3 × 2 × 1
Next, let's divide (2n)! by n!:
(2n)!/n! = [(2n) × (2n-1) × (2n-2) × ... × 3 × 2 × 1] / [n × (n-1) × (n-2) × ... × 3 × 2 × 1]
We can observe that all the terms in the denominator cancel out with some terms in the numerator:
(2n)!/n! = [ (2n) × (2n-1) × (2n-2) × ... × (n+1) × n × (n-1) × (n-2) × ... × 3 × 2 × 1] / [n × (n-1) × (n-2) × ... × 3 × 2 × 1]
This leaves us with:
(2n)!/n! = (2n) × (2n-1) × (2n-2) × ... × (n+1)
We can see that the resulting expression on the right side is exactly equal to 1 × 3 × 5 × ... × (2n-1) × 2n, as required.
Therefore, we have proven that (2n)!/n! = 1 × 3 × 5 × ... × (2n-1) × 2n.