Find the explicit formula for the following sequence 0, 1, 3, 6, 10 ...

Zachariah Norris

Zachariah Norris

Answered question

2022-09-24

Find the explicit formula for the following sequence 0, 1, 3, 6, 10 ...

Answer & Explanation

arribanti9

arribanti9

Beginner2022-09-25Added 5 answers

A finite sequence does not determine a unique formula, but in this particular case there are enough terms to see an intended pattern.
We can derive an explicit formula by examining sequences of differences.
The given sequence is:
0,1,3,6,10
Write down the sequence of differences of that sequence:
1,2,3,4
Write down the sequence of differences of the sequence:
1,1,1
Having arrived at a constant sequence, we can now use the initial term of each of these sequences as coefficients to construct an explicit polynomial formula:
a n = 0 0 ! + 1 1 ! ( n - 1 ) + 1 2 ! ( n - 1 ) ( n - 2 )
= n - 1 + 1 2 n 2 - 3 2 n + 1
= 1 2 n 2 - 1 2 n
= 1 2 n ( n - 1 )

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