The next term in the series is: 2, 5, 9, 19, 37, ?
75
A number series is solved by finding a single rule that carries each term to the next, so the first step is to see how consecutive terms are related.
The differences are 3, 4, 10, 18, which form no obvious pattern, so a purely additive rule can be set aside. Looking instead at ratios, each term is roughly double the previous one, which suggests a multiplicative rule with a small correction.
Testing that idea: 2 × 2 + 1 = 5; 5 × 2 − 1 = 9; 9 × 2 + 1 = 19; 19 × 2 − 1 = 37. The rule holds throughout - each term is doubled, and one is alternately added and subtracted.
Since the last operation applied was a subtraction, the next must be an addition: 37 × 2 + 1 = 75.
The distractors are placed to catch partial reasoning. Choosing 73 comes from applying the wrong sign, 37 × 2 − 1, and so from missing the alternation. The values 78 and 80 do not follow from any consistent doubling rule at all.
Hence the next term in the series is 75.
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