What is the $20^{\text {th }}$ term of the sequence defined by $$a_{n}=(n-1)(2-n)(3+n) ?$$

Step 1: Understand the Problem

The problem asks for the $20^{\text {th }}$ term of a sequence defined by a given formula. The formula for the $n^{\text {th }}$ term, $a_n$, is provided as a product of three factors involving $n$. To find the $20^{\text {th }}$ term, we need to substitute $n=20$ into this formula.

Step 2: Substitute $n = 20$ into the formula

To find the $20^{\text {th }}$ term, we replace every instance of $n$ in the given formula with the number $20$. This substitution will allow us to calculate the value of $a_{20}$.

Step 3: Calculate each factor

Now, we evaluate each of the three factors separately. The first factor is $20-1=19$. The second factor is $2-20=-18$. The third factor is $3+20=23$.

Step 4: Multiply the factors

Finally, we multiply the three calculated factors together to find the value of $a_{20}$. First, multiply $19 \times (-18)$, which gives $-342$. Then, multiply $-342 \times 23$.

Step 5: Final Calculation

Performing the multiplication $-342 \times 23$ yields $-7866$. This is the $20^{\text {th }}$ term of the sequence.

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