Choose the next term in the alternating +5, −7 pattern: 71, 76, 69, 74, 67, 72, ( ? )

Introduction / Context:This completion problem uses alternating fixed differences: add a constant, then subtract a (different) constant, repeatedly. Recognizing the two-step cycle is the key to predicting the next value.

Given Data / Assumptions:

• Sequence: 71, 76, 69, 74, 67, 72, ( ? )
• We suspect alternating +5 and −7.

Concept / Approach:Compute successive differences: 71→76 (+5), 76→69 (−7), 69→74 (+5), 74→67 (−7), 67→72 (+5). The pattern dictates the next step must be −7.

Step-by-Step Solution:After 67, +5 gives 72 (shown)Next move is −7 ⇒ 72 − 7 = 65

Verification / Alternative check:Replaying the pair (+5, −7) repeatedly rebuilds all terms and uniquely implies 65 next.

Why Other Options Are Wrong:77 and 76 correspond to +5 moves; 80 and 69 do not continue the established cycle.

Common Pitfalls:Assuming a single arithmetic progression; the two-step cycle is essential here.

Final Answer:65