A home-schooled student is analyzing a geometric sequence where the first term is 3 and the common ratio is 2. If the sum of the first $ n $ terms is 1023, what is $ n $? - Sterling Industries
A home-schooled student is analyzing a geometric sequence where the first term is 3 and the common ratio is 2. If the sum of the first $ n $ terms is 1023, what is $ n $?
A home-schooled student is analyzing a geometric sequence where the first term is 3 and the common ratio is 2. If the sum of the first $ n $ terms is 1023, what is $ n $?
Why this problem is resonating now
In a digital age where curiosity about patterns and logic thrives, students—and curious minds everywhere—are increasingly exploring real-world math through structured problems like geometric sequences. This particular question, featuring a first term of 3, a ratio of 2, and a total sum of 1023, reflects growing interest in foundational algebra concepts that support STEM thinking. As families seek meaningful, screen-based learning tools, problems rooted in sequences bridge theory with tangible reasoning—especially among home-schooled students who thrive on independent inquiry.
How this seqence analysis unfolds
Geometric sequences follow a clear pattern: each term is multiplied by a fixed number—the common ratio. Here, starting at 3 and multiplying by 2 yields:
3, 6, 12, 24, 48, 96, 192, 384, 768, 1536…
The sum $ S_n $ of the first $ n $ terms uses the formula:
$$
S_n = a \cdot \frac{r^n - 1}{r - 1}
$$
With $ a = 3 $, $ r = 2 $, this becomes:
$$
S_n = 3 \cdot \frac{2^n - 1}{2 - 1} = 3(2^n - 1)
$$
Setting this equal to 1023 gives:
$$
3(2^n - 1) = 1023
$$
Dividing both sides by 3:
$$
2^n
