What is the 20th term of the sequence defined by an=(n−1)(2−n)(3+n)? $
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Step-by-Step Solution
Step 1: Understand the Problem
The problem asks for the 20th term of a sequence defined by a given formula. The formula for the nth term, an, is provided as a product of three factors involving n. To find the 20th term, we need to substitute n=20 into this formula.
Step 2: Substitute n=20 into the formula
To find the 20th 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 a20.
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 a20. First, multiply 19×(−18), which gives −342. Then, multiply −342×23.
Step 5: Final Calculation
Performing the multiplication −342×23 yields −7866. This is the 20th term of the sequence.