Question

How do you find the $n^{th}$ term of the sequence 2, 5, 10, 17, 26, 37, …… ?

Answer 1

Here in this question, we have to find the explicit formula of the sequence. As we can see, we are not assigned with the number of terms in this sequence. So in that case, we need to just find the explicit formula in terms of ‘n’ where ‘n’ is the number of terms in the given sequence.

Complete step by step answer:
Let’s solve the question now.
When we start examining this sequence, we see that there is no common difference. The difference between the terms are: 3, 5, 7, 9, 11 and so on. In this way the series is increasing. And we have not assigned any specific number of terms. So we have to let that we have ‘n’ terms in our sequence. When we look at the first term, we just assume that it is formed by 1 + 1 which is equal to 2 that is our first term. Further, when to move to next term i.e. 5, just break it as 4 + 1 which can be written as ${{2}^{2}}+1$ also. Similarly when we move to 10, break it as 9 + 1 = ${{3}^{2}}+1$. For next term which is 17 it can be ${{4}^{2}}+1$. So this will go in a similar fashion. Now if we just write the sequence again and write it in a more expanded form, we will get:
$\Rightarrow$2, 5, 10, 17, 26, 37, ……
Now, expand every number of the sequence:
$\Rightarrow \left( {{1}^{2}}+1 \right)+\left( {{2}^{2}}+1 \right)+\left( {{3}^{2}}+1 \right)+\left( {{4}^{2}}+1 \right)+.......$
Here, we can observe that the square of a number keeps on increasing but the 1 is remaining constant. The base can increase up to ‘n’ number.
As we know that the explicit formula of an arithmetic sequence is designated to the $n^{th}$ term of the sequence. So the formula for the sequence in terms of ‘n’ will be:
$\Rightarrow {{n}^{2}}+1$
This is the final answer.

Note: There is no need to apply the formula: ${{a}_{n}}={{a}_{1}}+(n-1)d$ for finding the $n^{th}$ term because it won’t work for this sequence. This formula will only work if we have the common difference i.e. ‘d’. Before applying any formula we should know the first term i.e. ${{a}_{1}}$, secondly the common difference ‘d’ and thirdly the number of terms. So the first step is to examine the sequence.