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

floymdiT

Answered question

2021-09-19

Find a recurrence relation satisfied by this sequence.

a_{n} = n^{2} + n

Layton

Skilled2021-09-20Added 89 answers

Given:

a_{n} = n^{2} + n

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

a_{n}

by 0:

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

Let us similarly determine the next few terms as well:

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

a_{2} = 2^{2} + 2 = 6 = a_{1} + 2 (2)

a_{3} = 3^{2} + 3 = 12 = a_{2} + 2 (3)

a_{4} = 4^{2} + 4 = 20 = a_{3} + 2 (4)

a_{5} = 5^{2} + 5 = 30 = a_{4} + 2 (5)

a_{6} = 6^{2} + 6 = 42 = a_{5} + 2 (6)

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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