Solving the Number Series: Finding the Missing Term

This solution explains how to find the missing number in the sequence 27, 55, 112, 227, 458, ? using logical deduction and pattern recognition.

Analyzing the Number Sequence Pattern

The given number series is:

• 27
• 55
• 112
• 227
• 458
• ?

Step 1: Calculating Differences Between Consecutive Terms

To understand the pattern, let's calculate the difference between consecutive numbers in the series:

• First difference: \(55 - 27 = 28\)
• Second difference: \(112 - 55 = 57\)
• Third difference: \(227 - 112 = 115\)
• Fourth difference: \(458 - 227 = 231\)

The sequence of differences is 28, 57, 115, 231.

Step 2: Identifying the Pattern in the Differences

Now, let's examine the differences between these calculated differences (the second differences):

• First second difference: \(57 - 28 = 29\)
• Second second difference: \(115 - 57 = 58\)
• Third second difference: \(231 - 115 = 116\)

We notice a pattern in these second differences: each number is double the previous one.

• \(29 \times 2 = 58\)
• \(58 \times 2 = 116\)

Step 3: Predicting the Next Differences

Following this doubling pattern, the next second difference should be:

• Next second difference: \(116 \times 2 = 232\)

Now, we can find the next difference in the original sequence (the next first difference) by adding the predicted second difference to the last calculated first difference:

• Next first difference: \(231 + 232 = 463\)

Step 4: Calculating the Missing Number

Finally, to find the missing number in the original series, we add this next first difference to the last number in the given series:

• Missing Number: \(458 + 463 = 921\)

Alternative Pattern Verification

We can also find the pattern by looking at the operation between each term and the next term:

• \(27 \times 2 + 1 = 54 + 1 = 55\)
• \(55 \times 2 + 2 = 110 + 2 = 112\)
• \(112 \times 2 + 3 = 224 + 3 = 227\)
• \(227 \times 2 + 4 = 454 + 4 = 458\)

The pattern follows the rule: Multiply the previous term by 2 and add an increasing integer (\(+1, +2, +3, +4, \dots\)).

Applying this rule to find the next term:

• Next Term Calculation: \((458 \times 2) + 5 = 916 + 5 = 921\)

This confirms the result obtained using the difference method.

Conclusion

Based on the analysis of the differences and the alternative pattern verification, the missing number in the series 27, 55, 112, 227, 458, ? is 921.