Question

What comes next in the series?
$2,3,6,10,17,28,?$

Answer 1

Hint: The next term can be found only by knowing the pattern or the rule of the progression in the series. Start we finding a rule that satisfies all the terms. Here, notice that each term is the sum of the previous two-term plus one. Use this to find the next required term.

Complete step-by-step answer:
Here in this problem, we are given six numbers in the form of an increasing series, i.e. $2,3,6,10,17,28$ . With this information, we need to find the next term for this series.
We can find the seventh number in this series by first finding out the rule by which the progression from one term to its successive term takes place. The rules for the progression should be satisfied with all the elements of the series.
Here, we have the first term as $2$ and the successive term is given by: $2 + 0 + 1 = 3$
Then the second term is $3$ and the successive term can be given by: $3 + 2 + 1 = 6$
The third term in the series is $6$ and the successive term can be given by: $6 + 3 + 1 = 10$
Then again the fourth term in the series is $10$ and the successive term can be given by: $10 + 6 + 1 = 17$
So we get the fifth term of the series as $17$ and the successive term can be given as: $17 + 10 + 1 = 28$
Therefore, we found the rule of the progression in this series, and the same rule is satisfied by all the terms of the series.
So according to the rule of the progression for this given series $2,3,6,10,17,28$, we got that each term is obtained by adding up the previous two terms with $1$ , i.e. ${a_n} = {a_{n - 1}} + {a_{n - 2}} + 1$ , where $'{a_n}'$ is the nth term of the series.
$ \Rightarrow $ For the seventh term, $n = 7$
Therefore, by using the above-established rule, we get:
$ \Rightarrow {a_7} = {a_6} + {a_5} + 1$ or sum of the previous two terms plus $1$
Required next term $ = 17 + 28 + 1 = 46$
Thus, the series becomes $2,3,6,10,17,28,46$.
Note: In questions like this, finding the rule that satisfies the progression of all the terms is the most crucial part of the solution. An alternate approach can be found by taking the difference of the terms, i.e. $3 - 2 = 1,6 - 3 = 3,10 - 6 = 4,17 - 10 = 7,28 - 17 = 11$ . Now we can find a pattern in the differences of the terms. $1 + 3 = 4,3 + 4 = 7,4 + 7 = 11$ and the next term can be given by $7 + 11 = 18$, i.e. $17 + 18 = 46$ .