Question: A sequence of five real numbers forms a geometric progression. The sum of the first three terms is 21, and the sum of the last three terms is 84. Find the common ratio.

Sun, 03 May 2026 23:17:14 +0000

**Discover Hook: What patterns connect five numbers in a quiet mathematical rhythm?**
In a world where patterns shape our understanding of everything from music to finance, a subtle geometric sequence recently sparked curiosity among math learners and problem solvers. The question surface: five real numbers form a geometric progression, with the first three adding to 21 and the last three summing to 84. Many users now seek the hidden valid common ratio—not as a quick fix, but as part of understanding how sequences shape real-world logic. This isn’t just an equation; it’s a puzzle bridging algebra and insight—one that reveals how small changes in ratios ripple across entire sequences. --- ### Why This Question Is Gaining Attention in the US In today’s digital landscape, curiosity-driven learning thrives on systems that feel accessible but deliver depth—like geometric progressions. This specific problem taps into a growing interest in structured thinking and logical reasoning, whether for academic study, coding applications, or financial modeling. Recent data shows rising engagement with STEM-focused content on mobile platforms—especially among users aged 18–45 seeking intellectual practice. Asking “the common ratio” isn’t merely academic; it’s a gateway to stronger analytical habits. The pairing of familiar terms—real numbers, geometric sequence, three-term sums—makes the problem immediately relatable. This subtle approach avoids jargon, inviting readers to explore the rhythm of mathematics without pressure, driving both dwell time and trust.