Q. 28 - Calculus

Question

Simplify the quotients in Exercises 21–28 without using a calculator.

$\frac{1·3·5\cdots (2k-1)}{1·3·5\cdots (2k+1)}$

Step-by-Step Solution

Answer

The answer is $\frac{1}{\left(2k\right)(2k+1)}.$

Step 1. Given information.

The given expression is $\frac{1·3·5\cdots (2k-1)}{1·3·5\cdots (2k+1)}$.

Step 2. Value of the expression.

We know,

$$\begin{gathered} n !=n(n-1) ! \\ Therefore, \\ \frac{1·3·5·\dots (2k-1)}{1·3·5·\dots (2k+1)}=\frac{1·3·5·\dots (2k-1)}{1·3·5·\dots (2k-1)\left(2k\right)(2k+1)} \\ =\frac{1}{\left(2k\right)(2k+1)} \end{gathered}$$