# What is the missing number in the sequence 1, 5, 10, 16, 23, 31,____?(a) 37(b) 38(c) 39(d) 40(e) 41

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Hint: We start solving by calculating the difference between every two consecutive numbers of the sequence. Once we calculate the differences, we can see that these differences are in AP (Arithmetic Progression). Using this AP (Arithmetic Progression), we find the difference 31 and the next number in the sequence. Once we find the difference, we add it to 31 to get the next number in the sequence.

Complete step-by-step solution:
We have a sequence given in the problem as 1, 5, 10, 16, 23, 31……. We need to find the next number which is missing in the sequence.
We need to find the logic that is present between all these numbers. Let us find the difference between the consecutive numbers present in the sequence.
Now, we find the difference between 1 and 5 and let it be ${{d}_{1}}$.
So, we have ${{d}_{1}}=5-1$.
We have ${{d}_{1}}=4$ ----------(1).
Now, we find the difference between 5 and 10 and let it be ${{d}_{2}}$.
So, we have ${{d}_{2}}=10-5$.
We have ${{d}_{2}}=5$ -----------(2).
Now, we find the difference between 10 and 16 and let it be ${{d}_{3}}$.
So, we have ${{d}_{3}}=16-10$.
We have ${{d}_{3}}=6$ ----------(3).
Now, we find the difference between 16 and 23 and let it be ${{d}_{4}}$.
So, we have ${{d}_{4}}=23-16$.
We have ${{d}_{4}}=7$ ----------(4).
Now, we find the difference between 23 and 31 and let it be ${{d}_{5}}$.
So, we have ${{d}_{5}}=31-23$.
We have ${{d}_{5}}=8$ ----------(5).
From equations (1), (2), (3), (4) and (5), we can see that ${{d}_{1}}$, ${{d}_{2}}$, ${{d}_{3}}$, ${{d}_{4}}$ and ${{d}_{5}}$ are A.P (Arithmetic Progression) with a common difference of 1.
Let us assume that the next number in the sequence be ‘x’ and the difference between ‘x’ and 31 be ${{d}_{6}}$.
We can find ${{d}_{6}}$ by adding 1 to ${{d}_{5}}$.
So, ${{d}_{6}}=1+{{d}_{5}}$.
${{d}_{6}}=1+8$.
${{d}_{6}}=9$.
Now, the value of ‘x’ is $x=31+{{d}_{6}}$.
$x=31+9$.
$x=40$.
$\therefore$ The next number in the sequence is 40.
The correct option for the given problem is (d).

Note: We always need to check whether we get any relation between the differences of two consecutive terms of the sequence. We should not make any calculation errors while calculating the difference and next numbers. Similarly, we can expect to find the next three terms of sequence, the sum of the next three terms in the sequence, the product of the next three terms in the sequence.