Question

Determine the value of $\prod _{k=1}^{100} \frac{k+1}{k}$.(The notation used here for products is defined in the preamble to Exercise 43 in Section 2.4.)

Short answer

$\prod _{k=1}^{100} \frac{k+1}{k}=101$

Step-by-step solution

Step: 1

$\prod _{k=1}^{100} \frac{k+1}{k}$

$$\begin{gathered} \mathrm{K}=1,\prod _{k=1}^{100} \frac{1+1}{1}=2 \\ \mathrm{K}=2,\prod _{k=1}^{100} \frac{2+1}{2}=\frac{3}{2} \end{gathered}$$

Step: 2

$$\begin{gathered} \mathrm{K}=3,\prod _{k=1}^{100} \frac{3+1}{3}=\frac{4}{3} \\ \mathrm{K}=5,\prod _{k=1}^{100} \frac{5+1}{5}=\frac{6}{5} \end{gathered}$$

Step: 3

K=99,$\prod _{k=1}^{100} \frac{99+1}{99}=\frac{100}{99}$

K=100,$\prod _{k=1}^{100} \frac{100+1}{101}=\frac{101}{100}$

Now, the denominator of next term eliminates the numerator.

At the end, the product have only numerator of last term and denominator of 1.

$\prod _{k=1}^{100} \frac{k+1}{k}=101$