- A
- B
- C
- D
Step-by-Step Solution
. Confirm with the next gap: . A common difference may be a fraction.
Answer: A)
Find the common difference of the AP
Class 10Arithmetic ProgressionsAlgebra — Sequences
Key idea: Recognising an AP
A sequence is an AP iff the difference of consecutive terms is constant.
**Test**: Compute a₂ − a₁ and a₃ − a₂. If they're equal, it's likely an AP — verify with a₄ − a₃.
**Examples**:
- 4, 7, 10, 13: differences 3, 3, 3 → AP, d = 3.
- 1, 4, 9, 16: differences 3, 5, 7 → NOT an AP (these are squares).
- 5, 5, 5, 5: differences 0, 0, 0 → AP with d = 0 (a constant sequence is an AP).
- 2, 6, 12, 20: differences 4, 6, 8 → NOT an AP.
The common difference d can be positive (increasing AP), negative (decreasing AP), or zero (constant AP).
**Test**: Compute a₂ − a₁ and a₃ − a₂. If they're equal, it's likely an AP — verify with a₄ − a₃.
**Examples**:
- 4, 7, 10, 13: differences 3, 3, 3 → AP, d = 3.
- 1, 4, 9, 16: differences 3, 5, 7 → NOT an AP (these are squares).
- 5, 5, 5, 5: differences 0, 0, 0 → AP with d = 0 (a constant sequence is an AP).
- 2, 6, 12, 20: differences 4, 6, 8 → NOT an AP.
The common difference d can be positive (increasing AP), negative (decreasing AP), or zero (constant AP).
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