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Answer :
Final answer:
The missing number in the series 6, 30, ?, 180, 330, 546, and 840 is 192, determined by identifying the pattern of differences between the terms and the second differences.
Explanation:
The sequence given is 6, 30, ?, 180, 330, 546, and 840. To find the missing term, it's helpful to try to understand the pattern of the series. In this case, differences between each successive term increase by 24, suggesting a second difference series.
First, calculate the differences between each number: 30-6 = 24, 180-30 = 150, 330-180 = 150, 546-330 = 216, and 840-546 = 294. Now create a second sequence of differences from these numbers: 150-24 = 126, 150-150 = 0, 216-150 = 66, and 294-216 = 78.
It seems that each difference in this second series also increases by an increment of 12 each time. Therefore, the second difference between the missing number and 30 should be 126 + 12 = 138. So the difference between the missing number and 30 is 24 + 138 = 162. Therefore, the missing number is 30 + 162 = 192.
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