Expert answers to "Write a recursive rule for the sequence. 2,..."

khi1la2f1qv

Answered question

2021-11-11

Write a recursive rule for the sequence. 2, 2, 4, 12, 48, ....

Walker Funk

Beginner2021-11-12Added 13 answers

We can use a table to organize the terms. Consider table

\begin{array}{ll} n & 1 & 2 & 3 & 4 & 5 \\ a_{n} & 2 & 2 & 4 & 12 & 48 \end{array}

From table we can conclude that the terms do not have a common difference because

a_{4} - a_{3} = 8 \neq 0 = a_{2} - a_{1}

Also, the terms do not have a common ratio.
We can see that first is

a_{1} = 2

. Note that

a_{2} = 2 = 1 \cdot 2 = 1 \cdot a_{1}

a_{3} = 4 = 2 \cdot 2 = 2 \cdot a_{2}

a_{4} = 12 = 3 \cdot 4 = 3 \cdot a_{3}

a_{5} = 48 = 4 \cdot 12 = 4 \cdot a_{4}

Thus we can conclude that

a_{n} = (n - 1) \cdot a_{n - 1}

Therefore, a recursive rule for the sequence is

a_{1} = 2, a_{n} = (n - 1) \cdot a_{n - 1}

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