Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs

Dashboard
Notifications
Mark all as read
Q&A

ratio of partial sums of the same geometric sequence

+1
−0

My kid was given this question:

In a geometric sequence, the proportion of (the sum of the first $12$ terms) to (the sum of the first $8$ terms) is $\frac{819}{51}$. Find the common ratio of the sequence.

The only formula thus far covered for the partial sum of a geometric sequence $(a_1q_i)_{i\ge0}$ is$$\frac{a_1(q^n-1)}{q-1}.$$ How does one solve this?


The solution I found is as follows, but it seems too roundabout for the context, so I wonder what more direct way there is:

The given information means$$\frac{q^{12}-1}{q^8-1}=\frac{819}{51}.$$ Let $u=q^4$. Then$$\frac{819}{51}=\frac{u^3-1}{u^2-1}=\frac{u^2+u+1}{u+1}$$ $$\frac{819-51}{51}=\frac{u^2}{u+1}$$ $$51u^2-768u-768=0$$so $u\in\lbrace-16/17,16\rbrace$ and $q=\pm2$.

Why does this post require moderator attention?
You might want to add some details to your flag.
Why should this post be closed?

0 comment threads

1 answer

+1
−0

I think your approach is the expected one, but a shortcut if rigour is not required would be to note that the absolute value of the ratio must be greater than 1, or the proportion couldn't exceed $\frac{12}8$; but then the largest term dominates, so $$q^4 \approx \frac{819}{51} \approx 16.05$$ and then trial and error shows that $q^4 = 16$ works.

(Of course, this doesn't exclude other solutions: see "if rigour is not required" above).

Why does this post require moderator attention?
You might want to add some details to your flag.

0 comment threads

Sign up to answer this question »

This community is part of the Codidact network. We have other communities too — take a look!

You can also join us in chat!

Want to advertise this community? Use our templates!

Like what we're doing? Support us! Donate