To simplify the given expression [tex]\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\ldots\left(1+\frac{1}{20}\right)\)[/tex], let's proceed step by step.
Each term in the product can be written as:
[tex]\[
1 + \frac{1}{i} \quad \text{for} \quad i = 2, 3, 4, \ldots, 20
\][/tex]
This means:
[tex]\[
\left(1+\frac{1}{2}\right) \left(1+\frac{1}{3}\right) \left(1+\frac{1}{4}\right) \ldots \left(1+\frac{1}{20}\right)
\][/tex]
Converting these expressions:
[tex]\[
\left(1 + \frac{1}{2}\right) = \frac{3}{2}, \quad \left(1 + \frac{1}{3}\right) = \frac{4}{3}, \quad \left(1 + \frac{1}{4}\right) = \frac{5}{4}, \quad \ldots, \quad \left(1 + \frac{1}{20}\right) = \frac{21}{20}
\][/tex]
Thus, the product is:
[tex]\[
\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{21}{20}
\][/tex]
Notice that many terms in this product cancel out:
[tex]\[
\frac{3 \cdot 4 \cdot 5 \cdot 6 \cdot \ldots \cdot 21}{2 \cdot 3 \cdot 4 \cdot 5 \cdot \ldots \cdot 20}
\][/tex]
Almost all terms in the numerator and denominator cancel, leaving:
[tex]\[
\frac{21}{2}
\][/tex]
Therefore:
[tex]\[
\frac{21}{2} = 10.5
\][/tex]
So, the simplified result of the expression [tex]\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)\left(1+\frac{1}{4}\right)\ldots\left(1+\frac{1}{20}\right)\)[/tex] is indeed:
[tex]\[
10.5
\][/tex]
