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Question

Difficulty: MediumNumber and Alphabet Series

A cryptography algorithm generates a continuous stream of security tokens based on a polynomial progression. The first five tokens generated by the system are 55, 1212, 3131, 6868, and 129129.

What is the numerical value of the sixth token generated by this algorithm?

Answer: 220

Answer

220
The sequence follows the mathematical rule where the nn-th term is equal to n3+4n^3 + 4. The first term is 13+4=51^3 + 4 = 5, the second is 23+4=122^3 + 4 = 12, up to the fifth which is 53+4=1295^3 + 4 = 129. Applying this rule to the sixth position yields 63+4=216+4=2206^3 + 4 = 216 + 4 = 220. Alternatively, solving via successive differences confirms that the constant third difference is 6, which accurately points to 220.

Step-by-Step Solution

1
Calculate the first-level differences between the given consecutive tokens.
The sequence of differences is 7, 19, 37, and 61.
Establishing the initial rate of change helps identify if a linear or higher-order polynomial pattern exists.
2
Calculate the second-level differences from the results of Step 1.
The differences between the differences are 12, 18, and 24.
Finding the differences of the differences reveals simpler underlying arithmetic progressions.
3
Identify the pattern in the second-level differences and extrapolate the next value.
The values (12, 18, 24) increase by exactly 6 each time. The next second-level difference is 24 + 6 = 30.
Extending this constant third-level difference (+6) is required to build the sequence forward.
4
Calculate the next first-level difference and the final sequence term.
Next first-level difference = 61 + 30 = 91. Next sequence term = 129 + 91 = 220.
Applying the extrapolated values back up the chain yields the target term in the main sequence.

Key Concept

Number series completion using the method of successive differences or cube offsets.
Estimated Time:1m 15s
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Topics

Number and Alphabet Series

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