What should come in place of the question mark (?) in the given series? 22,…

Concept: In an alternating number series a single common difference does not govern every step. Two rules act in turn, so the differences between consecutive terms repeat in a fixed cycle, and the terms standing at alternate places form two separate arithmetic progressions. The working method is therefore to write out all consecutive differences first and identify the repeating cycle before predicting the next term.

Application: Take the consecutive differences of the given series 22, 21, 24, 23, 26, ? one step at a time.

1. 21 - 22 = -1

2. 24 - 21 = +3

3. 23 - 24 = -1

4. 26 - 23 = +3

5. The differences run -1, +3, -1, +3, so the repeating cycle is (-1, +3) and the step that follows a +3 must be a -1.

6. Applying that step to the last given term 26 gives 26 - 1 = 25, so the missing term is 25.

Cross-check: Split the series by alternate places instead of by steps.

• The terms 22, 24 and 26, which stand at alternate places, form an arithmetic progression with common difference +2.

• The remaining terms 21, 23 and the missing term form another arithmetic progression with the same common difference +2, so the missing term is 23 + 2 = 25.

Both routes give the same missing term, 25.