Hint: The difference between the terms of the series is in arithmetic progression and that difference is added to the last number.
The given series is: 22, 26, 29, 31,……..
If we observe the terms of the series, it is not in arithmetic, geometric or harmonic progression. However, if we consider the difference between the terms of the series we will get:
4 is the difference between first two terms (i.e. 26-24),
3 is the difference between first two terms (i.e. 29-26),
2 is the difference between the first two terms (i.e. 31-29).
Therefore, the difference between the terms is in A.P. We have 4,3,2… as our difference series. So, its next term is obviously 1. This means that the difference between the next two terms of the initial series is 1. So, if s is the next term of the initial series, we have:
\Rightarrow s - 31 = 1, \\
\Rightarrow s = 1 + 31, \\
\Rightarrow s = 32 \\
Therefore the next term of the series is 32.
Note: If we observe our difference series, it is in decreasing A.P. If we extend it, we’ll get:
So, after a certain stage its terms will be negative. Which means our initial series will start decreasing as we can see if we follow it:
Thus the series is first increasing and then decreasing.