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Difficulty: MediumNumber and Alphabet Series

A transportation company schedules its daily long-distance routes by increasing the total daily mileage according to a specific mathematical progression. Over the first five days of a new expansion phase, the daily total route distances recorded are 55, 1212, 2626, 4949, and 8383 miles respectively.

If this pattern continues strictly, what will be the total route distance in miles for the sixth day?

Answer: 130 miles

Answer

130
The sequence follows a double-difference pattern where the second-level differences form a simple arithmetic progression increasing by 22. By extending this pattern, the next addition to the sequence is 4747, which yields exactly 130130 when added to the fifth term.

Step-by-Step Solution

1
Calculate the differences between consecutive terms in the given sequence.
12−5=712 - 5 = 7; 26−12=1426 - 12 = 14; 49−26=2349 - 26 = 23; 83−49=3483 - 49 = 34.
Finding the first-level differences is the standard approach to identify hidden polynomial sequences.
2
Calculate the differences of the first-level differences (second-level differences).
14−7=714 - 7 = 7; 23−14=923 - 14 = 9; 34−23=1134 - 23 = 11.
The first-level differences do not form a constant sequence, but measuring their rate of change reveals an underlying arithmetic progression.
3
Identify the pattern in the second-level differences and extrapolate the next value.
The second-level differences (77, 99, 1111) increase by 22 each time. The next second-level difference will be 11+2=1311 + 2 = 13.
Establishing the core constant pattern allows for accurate forward prediction.
4
Calculate the next first-level difference and the final 6th term.
Next first-level difference = 34+13=4734 + 13 = 47. Sixth term = 83+47=13083 + 47 = 130.
Applying the extrapolated differences forward up the chain yields the final required value.

Key Concept

Double Difference Number Series
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Topics

Number and Alphabet Series

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