Find the next three terms in 2,5,10,17,26,...

Question

Maleah Lester · Accepted Answer

Examine the sequence of differences of this sequence and its sequence of differences to find the next three terms, 37,50,65 and the general formula

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

To find a pattern in this sequence, first write out the original sequence:
(i) 2,5,10,17,26
Then write out the sequence of differences between successive terms of that sequence:
(ii) 3,5,7,9
Then write out the sequence of differences of that sequence:
(iii) 2,2,2
Having reached a constant sequence, there are a couple of things we can do:
(1) We can find the next three elements of the original sequence as requested.
To do this, add three more 2's to the last sequence:
2,2,2,2,2,2
Then add three more terms to the previous sequence using the three new elements of this sequence as differences:
3,5,7,9,11,13,15
Then add three more terms to the original sequence using the three new elements of this sequences as differences:
2,5,10,17,26,37,50,65
(2) We can find a general formula for then nth term

a_{n}

of the sequence by using the initial terms 2, 3, 2 of the sequences (i), (ii) and (iii) as coefficients:

a_{n} = \frac{2}{0!} + \frac{3}{1!} (n - 1) + \frac{2}{2!} (n - 1) (n - 2)

= 2 + 3 n - 3 + n^{2} - 3 n + 2 = n^{2} + 1

Find the next three terms in 2,5,10,17,26,...

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