Question

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

Answer 1

Hint: In order to find the next three terms in the sequence, we need to detect the pattern through which the sequence is moving forward. So, to detect the pattern we will look at the differences between every two successive terms in the given sequence, if we get a new sequence of the differences, we will again calculate the differences between every two successive terms in that sequence. We will continue this till we get a constant sequence and then we will add the next three terms according to the pattern detected.

Complete step-by-step answer:
(i)
When we are given a series of numbers and we have to figure out the pattern to write the next terms, our first attempt should be to find out the difference between the consecutive terms.
Since, we are given:
 $ 2,5,10,17,26 $
Therefore, we will find the difference between each consecutive term, i.e.,
 $
  5 - 2 = 3 \\
 10 -5 = 5 \\
 17 -10 = 7 \\
 26 -17 = 9 \;
$
(ii)
Now, we have got a sequence of differences between the successive terms of the original sequence given to us as:
 $ 3,5,7,9 $
Now, in order to find the pattern in this sequence, we will calculate the difference between the consecutive terms in this series. Therefore,
 $
  5 - 3 = 2 \\
  7 - 5 = 2 \\
  9 - 7 = 2 \;
  $
Therefore, this time the sequence of the differences between the successive terms of the sequence of the difference between the successive terms of the original sequence is:
 $ 2,2,2 $
As we can see that here the difference is equal i.e., it is a constant sequence, we can find the next three terms of the original sequence as asked by following the pattern.
(iii)
Therefore, first we will add three more $ 2 $ ’s to the last sequence. We will get:
 $ 2,2,2,2,2,2 $
Then we will add three more terms to the previous sequence using the three new elements of this sequence as differences:
 $ 3,5,7,9,\left( {9 + 2} \right),\left( {\left( {9 + 2} \right) + 2} \right),\left( {\left( {\left( {9 + 2} \right) + 2} \right) + 2} \right) $
Which on simplifying, becomes:
 $ 3,5,7,9,11,13,15 $
And now, we will add three more terms to the original sequence using the three new elements of this sequences as differences:
 $ 2,5,10,17,26,\left( {26 + 11} \right),\left( {\left( {26 + 11} \right) + 13} \right),\left( {\left( {\left( {26 + 11} \right) + 13} \right) + 15} \right) $
Which on simplifying, becomes:
 $ 2,5,10,17,26,37,50,65 $
Hence the next three terms in the sequence $ 2,5,10,17,26 $ are $ 37,50,65 $
So, the correct answer is “ $ 37,50,65 $ ”.

Note: After completing the second step of the solution i.e., writing down the sequence of the differences between the successive terms of the sequence which itself is created by the differences between the successive terms of the original sequence, we could also find a general formula for the $ n $ th term $ {a_n} $ of the sequence by using the initial terms of the first, second and the third sequence i.e., $ 2,3,2 $ as coefficients:
 $
  {a_n} = \dfrac{2}{{0!}} + \dfrac{3}{{1!}}\left( {n - 1} \right) + \dfrac{2}{{2!}}\left( {n - 1} \right)\left( {n - 2} \right) \\
  {a_n} = 2 + 3n - 3 + {n^2} - 3n + 2 \\
  {a_n} = {n^2} + 1 \;
  $