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Difficulty: MediumNumber and Alphabet Series

Analyze the numerical progression below to determine the rule governing the sequence. What is the value of the missing term that completes the series?

Answer:Consider the series: 7,16,41,90,171,【292】7, 16, 41, 90, 171, 【292】

Answer

The missing term that completes the sequence is 292.
The difference between successive terms follows the sequence of squares of consecutive odd numbers: 32=93^2 = 9, 52=255^2 = 25, 72=497^2 = 49, 92=819^2 = 81, and 112=12111^2 = 121. Adding 121121 to 171171 gives the missing term 292292.

Step-by-Step Solution

1
Calculate the differences between consecutive terms in the series.
16−7=916 - 7 = 9, 41−16=2541 - 16 = 25, 90−41=4990 - 41 = 49, 171−90=81171 - 90 = 81
Examining first-order differences helps uncover the underlying pattern.
2
Identify the mathematical relationship between the difference values.
The differences are 9,25,49,819, 25, 49, 81, which correspond to 32,52,72,923^2, 5^2, 7^2, 9^2.
The increments between consecutive terms represent the squares of consecutive odd integers starting from 3.
3
Calculate the next term in the sequence using the identified rule.
The next odd number after 99 is 1111, so the required increment is 112=12111^2 = 121. Adding this increment to the last given term gives 171+121=292171 + 121 = 292.
Applying the pattern's next step yields the correct missing term.

Key Concept

Series Completion using Differences of Squares of Consecutive Odd Numbers
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Topics

Number and Alphabet Series

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