Answer :
To determine the first five terms of the sequence defined by the formula [tex]\(a_n = 2 \cdot n!\)[/tex], we need to calculate each term individually by substituting values of [tex]\(n\)[/tex] from 1 to 5 into the formula.
Here's the step-by-step calculation for each term:
1. For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 2 \cdot 1! = 2 \cdot 1 = 2 \][/tex]
2. For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 2 \cdot 2! = 2 \cdot 2 = 4 \][/tex]
3. For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 2 \cdot 3! = 2 \cdot (3 \cdot 2 \cdot 1) = 2 \cdot 6 = 12 \][/tex]
4. For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 2 \cdot 4! = 2 \cdot (4 \cdot 3 \cdot 2 \cdot 1) = 2 \cdot 24 = 48 \][/tex]
5. For [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 2 \cdot 5! = 2 \cdot (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) = 2 \cdot 120 = 240 \][/tex]
Putting it all together, the first five terms of the sequence [tex]\(a_n = 2 \cdot n!\)[/tex] are:
[tex]\[ 2, 4, 12, 48, 240 \][/tex]
Therefore, the correct answer from the provided options is:
[tex]\[ 2, 4, 12, 48, 240 \][/tex]
Here's the step-by-step calculation for each term:
1. For [tex]\(n = 1\)[/tex]:
[tex]\[ a_1 = 2 \cdot 1! = 2 \cdot 1 = 2 \][/tex]
2. For [tex]\(n = 2\)[/tex]:
[tex]\[ a_2 = 2 \cdot 2! = 2 \cdot 2 = 4 \][/tex]
3. For [tex]\(n = 3\)[/tex]:
[tex]\[ a_3 = 2 \cdot 3! = 2 \cdot (3 \cdot 2 \cdot 1) = 2 \cdot 6 = 12 \][/tex]
4. For [tex]\(n = 4\)[/tex]:
[tex]\[ a_4 = 2 \cdot 4! = 2 \cdot (4 \cdot 3 \cdot 2 \cdot 1) = 2 \cdot 24 = 48 \][/tex]
5. For [tex]\(n = 5\)[/tex]:
[tex]\[ a_5 = 2 \cdot 5! = 2 \cdot (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) = 2 \cdot 120 = 240 \][/tex]
Putting it all together, the first five terms of the sequence [tex]\(a_n = 2 \cdot n!\)[/tex] are:
[tex]\[ 2, 4, 12, 48, 240 \][/tex]
Therefore, the correct answer from the provided options is:
[tex]\[ 2, 4, 12, 48, 240 \][/tex]
