Question: "Find a recurrence relation satisfied by this sequence...."

Emily-Jane Bray

Answered question

2021-09-16

Find a recurrence relation satisfied by this sequence.

a_{n} = n^{2}

Clara Reese

Skilled2021-09-17Added 120 answers

Given:

a_{n} = n^{2}

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

a_{n}

by 0:

a_{0} = 0^{2} = 0

Let us similarly determine the next few terms as well:

a_{1} = 1^{2} = 1 = a_{0} + 2 (1) - 1

a_{2} = 2^{2} = 4 = a_{1} + 2 (2) - 1

a_{3} = 3^{2} = 9 = a_{2} + 2 (3) - 1

a_{4} = 4^{2} = 16 = a_{3} + 2 (4) - 1

a_{5} = 5^{2} = 25 = a_{4} + 2 (5) - 1

a_{6} = 6^{2} = 36 = a_{5} + 2 (6) - 1

We note that each term is the previous term increased by 2n-1:

a_{n} = a_{n - 1} + 2 n - 1

Thus a recurrence relation for

a_{n}

is then:

a_{0} = 0

a_{n} = a_{n - 1} + 2 n - 1

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

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