Simple Arithmetic Sequence (Constant 1st Difference)
Core Pattern: In an arithmetic sequence, the difference between consecutive terms is constant ($d = x_n - x_{n-1}$). To find the next number, calculate the first difference $\Delta^{(1)}$ and add it to the last term.
Mathematical Formulation & Recurrence
\(x_n = x_1 + (n - 1)d \quad \iff \quad x_n = x_{n-1} + d\)
Method of Differences & Calculation Technique
The 1st Difference Method: Take adjacent pairs of numbers and subtract the earlier number from the later number (\(\Delta = x_{k+1} - x_k\)). If this difference is identical across all steps, the sequence is arithmetic. To find the next number: \(x_{\text{next}} = x_{\text{last}} + d\).
Visual Sequence & Difference Mapping
How to Solve This Series in 3 Steps:
- 1 Find the common step: Subtract the first term from the second (e.g. 11 - 7 = 4).
- 2 Verify consistency: Check that the same step holds for subsequent pairs (e.g. 15 - 11 = 4, 19 - 15 = 4).
- 3 Project the next term: Add the constant step $d$ (which may be negative!) to the last term to find the missing value.
Step-by-Step Worked Examples:
4, 11, 18, 25, 32, ?
Example 1
1. Differences: 11-4=7, 18-11=7, 25-18=7
2. Common difference d = +7
3. Next term: 32 + 7 = 39
Answer: 39
85, 76, 67, 58, 49, ?
Example 2
1. Differences: 76-85 = -9, 67-76 = -9
2. Negative difference d = -9
3. Next term: 49 - 9 = 40
Answer: 40
-15, -9, -3, 3, 9, ?
Example 3
1. Differences: -9 - (-15) = +6, 3 - (-3) = +6
2. Common difference d = +6
3. Next term: 9 + 6 = 15
Answer: 15
103, 117, 131, 145, ?
Example 4
1. Differences: 117-103=14, 131-117=14
2. Common difference d = +14
3. Next term: 145 + 14 = 159
Answer: 159
Interactive Comprehension Test
Instant Check
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7, 15, 23, 31, 39, ?
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