# How do you find the next three terms in 2,5,10,17,26,....?

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Examine the sequence of differences of this sequence and its sequence of differences to find the next three terms,

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To find a pattern in this sequence, first write out the original sequence:

(i)

#color(blue)(2),5,10,17,26#

Then write out the sequence of differences between successive terms of that sequence:

(ii)

#color(blue)(3),5,7,9#

Then write out the sequence of differences of that sequence:

(iii)

#color(blue)(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

#color(blue)(2),2,2,color(red)(2),color(red)(2),color(red)(2)#

Then add three more terms to the previous sequence using the three new elements of this sequence as differences:

#color(blue)(3),5,7,9,color(red)(11),color(red)(13),color(red)(15)#

Then add three more terms to the original sequence using the three new elements of this sequences as differences:

#color(blue)(2),5,10,17,26,color(red)(37),color(red)(50),color(red)(65)#

(2) We can find a general formula for then

#a_n = color(blue)(2)/(0!) + color(blue)(3)/(1!) (n-1) + color(blue)(2)/(2!) (n-1)(n-2)#

#=2+3n-3+n^2-3n+2 = n^2+1#

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#### Explanation:

Looking at the sequence it should be clear that it is neither arithmetic nor geometric.

Look at the differences:

,

This pattern is a clue that squares are involved.

Look at the pattern of the differences:

Comparing the square numbers with the given sequence, we can see that:

and so on to get the next terms as

Describe your changes (optional) 200