In the sequence of numbers 1, 2, 6, 33, 49, 174, 210, x, ......., the term x is:
553
To find the missing term \(x\) in the sequence: 1, 2, 6, 33, 49, 174, 210, \(x\), we need to identify a pattern or rule that defines the sequence. Let's examine the differences between consecutive terms:
- First, find the differences between consecutive terms:
- \(2 - 1 = 1\)
- \(6 - 2 = 4\)
- \(33 - 6 = 27\)
- \(49 - 33 = 16\)
- \(174 - 49 = 125\)
- \(210 - 174 = 36\)
- The differences are: 1, 4, 27, 16, 125, 36.
- These differences themselves appear to follow a recognizable pattern of \(n^3\) and \(n^2\). Observe:
- \(1 = 1^3\)
- \(4 = 2^2\)
- \(27 = 3^3\)
- \(16 = 4^2\)
- \(125 = 5^3\)
- \(36 = 6^2\)
- So, it seems the differences alternate between perfect cubes and perfect squares: \(1^3, 2^2, 3^3, 4^2, 5^3, 6^2\).
- The next difference in this alternating sequence should correspond to \(7^3\), which is 343.
- Adding this difference to the last known term \(210\), we get:
Therefore, the missing term \(x\) is \(\boxed{553}\), confirming option \(553\) is correct.
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