Reasoning topic
A number series is built by a rule, from one term to the next or across alternate terms. Write down what happens between the terms, and the rule is there to read.
A typical question asks for the term that fits the blank. Other questions ask for the next term, or for the one term that breaks the pattern.
One term is missing from the series below. What is it? 40, 46, 52, ?, 64, 70, 76
58
The gaps are all 6: 52 + 6 = 58, and 58 + 6 = 64.
Write the gap between each pair of neighbouring terms.
If the gaps are not steady, read them as a series of their own: their gaps, or a run you know, such as squares. If the terms grow or shrink fast, compare neighbours by division instead: the larger one divided by the smaller.
If the terms swing up and down, read the odd places and the even places as two separate series.
One term is missing from the series below. What is it? 5, 6, ?, 19, 35, 60, 96
Answer: 10
1. Look at the series below. One of the terms is incorrect. Which one? 4, 9, 14, 25, 36, 49
Do the terms look like a run of numbers you know?
14
The terms are squares: 2² = 4, 3² = 9, 4² = 16, 5² = 25, 6² = 36, 7² = 49. The third term should be 16, so 14 is the incorrect one.
2. Fill in the blank to complete the series: 10, 15, 13, ?, 16, 27, 19, 33, 22
The terms swing up and down. Read the odd places and the even places as two series.
21
Odd places: 10, 13, 16, 19, 22, rising by 3. Even places: 15, ?, 27, 33, rising by 6. The blank is in the 4th place, an even one, so it is 15 + 6 = 21. Check: 27 − 21 = 6.
3. One term is missing from the series below. What is it? 117649, 16807, ?, 343, 49, 7
The terms fall fast. Divide one term by the next.
2401
49 ÷ 7 = 7 and 343 ÷ 49 = 7, so each term is 7 times the one after it. 16807 ÷ 7 = 2401, and 2401 ÷ 7 = 343 agrees. The series is 7⁶, 7⁵, 7⁴, 7³, 7², 7¹.
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