Introduction / Context:
This problem hides a repeating three-step structure using multiples of 10 with magnitudes that grow from group to group. Recognizing grouped differences is key to extending the series accurately to the next two terms.
Given Data / Assumptions:
- Sequence: 75, 65, 85, 55, 45, 85, 35
- We need the next two numbers (positions 8 and 9).
- Differences appear to follow a recurring pattern in groups of three.
Concept / Approach:
Compute consecutive differences and look for repeating motifs such as [−10, +20, −30], followed by [−10, +40, −50], etc. When a pattern repeats with increasing magnitudes, project the next group to find the required terms.
Step-by-Step Solution:
Differences: 75→65 = −10; 65→85 = +20; 85→55 = −30; 55→45 = −10; 45→85 = +40; 85→35 = −50.We see grouped patterns: [−10, +20, −30] then [−10, +40, −50]. The first element of each group is always −10; the positive jump doubles each group (+20 → +40 → next +60); the large negative also grows by −20 (−30 → −50 → next −70).After 35, the next step (start of a new group) should be −10: 35 − 10 = 25.The following step should be +60: 25 + 60 = 85.Therefore, the next two terms are 25 and 85.
Verification / Alternative check:
Continuing one more step would be the group’s third jump −70: 85 − 70 = 15, which fits smoothly after 25, 85.
Why Other Options Are Wrong:
25 15 / 35 25 / 25 75 / 85 35: Either they break the −10, +even-multiple, increasing-magnitude rule or misorder the group structure.
Common Pitfalls:
Treating the series as a single arithmetic progression and missing the three-step grouped pattern with growing magnitudes.
Final Answer:
25 85
Discussion & Comments