Question

Write the first five terms of the sequence in which the nth term is a subscript n baseline equals Start Fraction left-parenthesis n plus 1 right-parenthesis exclamation-mark over n plus 1 End Fraction.
(1 point)
Responses

1, 2, 6, 24, 120
1, 2, 6, 24, 120

1, 2, 3, 4, 5
1, 2, 3, 4, 5

1, 1, 3, 12, 60
1, 1, 3, 12, 60

1, 1, 1, 1, 1

Answer 1

To find the first five terms of the sequence defined by the \( n \)-th term \( a_n = \frac{(n+1)!}{n+1} \), we first simplify this expression.

The factorial \( (n+1)! \) can be written as \( (n+1) \cdot n! \). So when we divide by \( n+1 \), we get:

\[
a_n = \frac{(n+1) \cdot n!}{n+1} = n!
\]

Now we can calculate the first five terms:

1. For \( n = 1 \): \( a_1 = 1! = 1 \)
2. For \( n = 2 \): \( a_2 = 2! = 2 \)
3. For \( n = 3 \): \( a_3 = 3! = 6 \)
4. For \( n = 4 \): \( a_4 = 4! = 24 \)
5. For \( n = 5 \): \( a_5 = 5! = 120 \)

Thus, the first five terms of the sequence are:

1, 2, 6, 24, 120

So the correct response is:

**1, 2, 6, 24, 120**