6, 10, 34, 178, 1042, ?
Understand the Problem
The question presents a sequence of numbers and asks for the next number in the series. To solve it, we will need to identify the pattern or rule governing the progression of numbers.
Answer
The next number in the sequence is $6226$.
Answer for screen readers
The next number in the sequence is $6226$.
Steps to Solve
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Identify the pattern in differences Calculate the differences between consecutive terms:
- (10 - 6 = 4)
- (34 - 10 = 24)
- (178 - 34 = 144)
- (1042 - 178 = 864)
So the differences are: (4, 24, 144, 864).
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Calculate the ratios of differences Now let's find the pattern in these differences:
- ( \frac{24}{4} = 6)
- ( \frac{144}{24} = 6)
- ( \frac{864}{144} = 6)
The ratio of differences is consistent (each is multiplied by 6).
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Find the next difference To find the next difference, multiply the last difference by 6: $$ 864 \times 6 = 5184 $$
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Calculate the next term To find the next number in the sequence, add the next difference to the last term: $$ 1042 + 5184 = 6226 $$
The next number in the sequence is $6226$.
More Information
The sequence is generated by a pattern of differences that are multiplied by a constant. This shows how sequences can often exhibit exponential growth through consistent multiplication.
Tips
- Ignoring the differences: Sometimes, just looking at the numbers may lead to overlooking the pattern in their differences.
- Incorrectly calculating ratios: Ensure that division is performed correctly and consistently for proper pattern recognition.
- Forgetting to follow through the calculated differences: Always check if the series is continued logically by validating through calculations of added differences.
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