Math Problem Statement

6,8,18,48,116,?

Solution

The sequence 6,8,18,48,116,?6, 8, 18, 48, 116, \,? appears to follow a pattern where each term is generated based on the previous ones. Let's analyze the differences between consecutive terms:

  1. 86=28 - 6 = 2
  2. 188=1018 - 8 = 10
  3. 4818=3048 - 18 = 30
  4. 11648=68116 - 48 = 68

Now, observe the differences between the differences:

  1. 102=810 - 2 = 8
  2. 3010=2030 - 10 = 20
  3. 6830=3868 - 30 = 38

These second differences aren't consistent, but let's look at the third differences:

  1. 208=1220 - 8 = 12
  2. 3820=1838 - 20 = 18

The third differences show a pattern, suggesting that the next third difference might be 18+6=2418 + 6 = 24. If we apply this backward:

  1. The next second difference would be 38+24=6238 + 24 = 62.
  2. The next first difference would be 68+62=13068 + 62 = 130.

So, the next term in the sequence would be:

116+130=246116 + 130 = 246

Thus, the next term in the sequence is 246.

Would you like more details, or do you have any questions? Here are five related questions you might find interesting:

  1. What is the significance of identifying patterns in sequences?
  2. How can we determine the general formula for a sequence?
  3. How do arithmetic and geometric sequences differ?
  4. Can you identify the pattern in a different sequence of your choice?
  5. What are some real-life applications of recognizing numerical patterns?

Tip: When analyzing a sequence, consider looking at the differences between terms, and if necessary, the differences of those differences, to uncover hidden patterns.

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Math Problem Analysis

Mathematical Concepts

Sequence
Pattern Recognition
Arithmetic Sequences

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12