Complete the series given below by finding the missing term. 1, 2, 9, ?, 65, 126
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28
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82
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99
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108
The series is n^3 + 1: 0^3+1=1, 1^3+1=2, 2^3+1=9, 3^3+1=28, 4^3+1=65, 5^3+1=126. The missing term is 3^3+1 = 28.
The series follows the pattern of perfect cubes with alternating signs: 1 cubed with a zero exponent is 1, 2 cubed minus 1 is 8 plus 1 equals 9, and so forth, but a simpler view is the sequence n cubed plus 1 for even positions and n cubed minus 1 for odd positions. For the fourth term, calculating 4 cubed minus 1 gives 64 minus 1 = 63, which does not match the given options, so let us look at the differences between terms which are 1, 7, 36, and 61; these represent 1 cubed, and the pattern of adding consecutive differences reveals the missing term is 28.