Question

Find the next two terms in the sequence:
\[1,2,4,7,11,...\]

Answer 1

Hint: We are given a sequence and based on the given sequence, we have to find the next two terms of the sequence. If we observe the sequence, we can see that the difference between the terms increases by 1 unit as the sequence proceeds, that is, from 1 to 2 the difference is 1. Then, from 2 to 4, the difference is 2 and then from 4 to 7, the difference is 3 and it increases so on. In order to find the next two terms, we will have to add the appropriate difference to the previous number to get the new number. Hence, we will have the required terms.

Complete step by step answer:
According to the given question, we are given a sequence and based on the characteristics of the given sequence we have to find the next two terms of the sequence.
The sequence that we have is,
\[1,2,4,7,11,...\]
We can label the terms as,
\[{{a}_{1}}=1\], \[{{a}_{2}}=2\], \[{{a}_{3}}=4\], \[{{a}_{4}}=7\] and \[{{a}_{5}}=11\]
If we carefully observe the sequence, there is progression in the difference of the two consecutive numbers. The difference of the two consecutive numbers is as follows,
\[{{a}_{2}}-{{a}_{1}}=2-1=1\]
\[{{a}_{3}}-{{a}_{2}}=4-2=2\]
\[{{a}_{4}}-{{a}_{3}}=7-4=3\]
\[{{a}_{5}}-{{a}_{4}}=11-7=4\]
So, we can see that the difference in the numbers is increasing as the sequence proceeds forward in the order 1, 2, 3, 4, 5, 6 and so on. So, the next term of the sequence can be found by adding the appropriate difference to the last number of the sequence. So, we have,
\[{{a}_{6}}={{a}_{5}}+5=11+5=16\]
and \[{{a}_{7}}={{a}_{6}}+6=16+6=22\]
The next two terms of the sequence are 16 and 22.
The sequence now looks like,
\[1,2,4,7,11,16,22\]

Note: The sequence should be correctly understood in order to be able to find the next two terms of the sequence. The factor that should be added to the last number in order to get the new term should be carefully calculated else the entire sequence will get wrong.
There is another way to know the factor that is to be added to the last number to get the new number of the sequence. If we want to find the sixth term, then we will add ‘6 – 1’ to the fifth term. And for the seventh term, we will add ‘7 – 1’ to the sixth term. That is, in general we can write the factor to be added for the nth term is \[n-1\].