Finding the wrong term

Learning outcome

You can find the one term that breaks a rule which fits the rest of the series.

Concepts and reasons

A wrong-term question still has one rule. Build that rule from the longest run that agrees, then see which single term refuses it.

Start at the left. Suppose the rule is times 2, plus 2. Then 4 becomes 10, 10 becomes 22, 22 becomes 46, and 46 becomes 94. If the printed series says 96 in that place, 96 is the wrong term. Check the term after it as well: 94 times 2, plus 2, is 190. The printed 190 agrees, so only 96 is out.

Do not mark the last term wrong merely because you have not found a rule yet. If no single rule fits any long run, look again. Guessing the last term is not a method.

Worked examples

1. Series: 4, 10, 22, 46, 96, 190. Rule: times 2, plus 2. 46 should be followed by 94, not 96. The wrong term is 96.
2. Series: 2, 5, 10, 17, 26, 38. Gaps of the correct series are +3, +5, +7, +9, +11. From 26 the next term is 37, not 38. The wrong term is 38.
3. Series: 3, 6, 12, 24, 48. Every step is times 2. There is no wrong term in this line. If you cannot find one, recheck the gaps before you pick an end term.

Common traps

Changing the rule halfway, so that every term can be forced to fit, hides the break. One early gap can look small. In 2, 5, 10, 17, 26 the gaps 3, 5, 7, 9 already tell you the next gap must be 11.

Practice questions

1. Which term is wrong: 5, 11, 23, 47, 96?
2. Which term is wrong: 1, 4, 9, 16, 24, 36?
3. Which term is wrong: 8, 27, 64, 125, 216?

Answers and explanations

1. 96. The rule is times 2, plus 1. 47 times 2, plus 1, is 95, not 96.
2. 24. The terms are squares of 1, 2, 3, 4, 5 and 6. 5 squared is 25, not 24. 36 is 6 squared and is correct.
3. None of these terms is wrong. They are the cubes of 2, 3, 4, 5 and 6. Do not discard 216 just because it is last.

Analogy

A wrong term is a cracked step in a staircase that otherwise rises by a rule. Walk from the bottom. The cracked step is the one that does not match the rise you have already measured.

Quick reference

• Build the rule from the longest agreeing run.
• Check the term after the suspect as well as the term before it.
• Times 2 plus 2: 4, 10, 22, 46, 94, 190. The intruder 96 stands where 94 belongs.
• Squares: 1, 4, 9, 16, 25, 36. The intruder 24 stands where 25 belongs.
• If every term fits, there is no wrong term.