Question

# Find the next term of the series:22, 26, 29, 31,â€¦â€¦..(A) 32 (B) 33 (C) 34 (D) 35

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,â€¦â€¦..
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 4,3,2â€¦ 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, weâ€™ll get:
4,3,2,1,0,-1,â€¦.
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:
22,26,29,31,32,32,31,29â€¦..
Thus the series is first increasing and then decreasing.