To analyze the sequence and determine the next term, let's look at the pattern present in the given numbers: [tex]\(14, 34, 62, 98\)[/tex].
### Step 1: Calculate the Differences Between Consecutive Terms
We start by finding the first differences between the consecutive terms:
- [tex]\(34 - 14 = 20\)[/tex]
- [tex]\(62 - 34 = 28\)[/tex]
- [tex]\(98 - 62 = 36\)[/tex]
So, the first differences are [tex]\(20, 28, 36\)[/tex].
### Step 2: Calculate the Differences of the Differences
Next, we find the differences between the first differences to see if there is a pattern:
- [tex]\(28 - 20 = 8\)[/tex]
- [tex]\(36 - 28 = 8\)[/tex]
So, the second differences are [tex]\(8, 8\)[/tex].
### Step 3: Identify the Pattern
Since the second differences are constant ([tex]\(8\)[/tex]), it suggests that the sequence follows a quadratic pattern. In sequences, constant second differences typically imply a quadratic relationship.
### Step 4: Use the Pattern to Determine the Next Term
Given that the second differences are constant, we can use this information to predict the next term in the sequence:
1. The last first difference is 36. If we add the constant second difference (8) to this, we get:
[tex]\[
36 + 8 = 44
\][/tex]
2. To find the next term in the original sequence, we add this new first difference (44) to the last term (98):
[tex]\[
98 + 44 = 142
\][/tex]
### Conclusion
The next term in the sequence [tex]\(14, 34, 62, 98\)[/tex] is [tex]\(142\)[/tex].
Thus, the sequence continued is [tex]\(14, 34, 62, 98, 142\)[/tex].
