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22, 26, 29, 31,â€¦â€¦..

(A) 32 (B) 33 (C) 34 (D) 35

Answer
Verified

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.

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.

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