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fortdefruitI

Answered question

2021-09-06

Find a recurrence relation satisfied by this sequence.

a_{n} = n!

Yusuf Keller

Skilled2021-09-07Added 90 answers

Given:

a_{n} = n!

Let us first determine the first term by replacing n in the given expression for

a_{n}

by 0:

a_{0} = 0 \neq 1

Let us first determine the first term by replacing n in the given expression for

a_{n}

by 0:

a_{0} = 0 \neq 1

Let us similarly determine the next few terms as well:

a_{1} = 1 \neq 1 = 1 a_{0}

a_{2} = 2 \neq 2 = 2 a_{1}

a_{3} = 3 \neq 6 = 3 a_{2}

a_{4} = 4 \neq 24 = 4 a_{3}

a_{5} = 5 \neq 120 = 5 a_{4}

a_{6} = 6 \neq 720 = 6 a_{5}

We note that each term is the previous term multiplied by n:

a_{n} = n a_{n - 1}

Thus a recurrence relation for

a_{n}

is then:

a_{0} = 1

a_{n} = n a_{n - 1}

Note: There are infinitely many different recurence relations that satisfy any sequence.

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