Arithmetic and Geometric Series Problem

[TheRedDevil18]

Homework Statement

The sum of the first 9 terms of an arithmetic series is 216. The first, third and seventh terms of the series form the first three terms of a Geometric pattern. Find the first term and the common difference of the arithmetic pattern.

Homework Equations

The Attempt at a Solution

S9 = 9/2(2a+8d) = 216
S3 = T1+T3+T7

Completely stuck and need help

 

[mfb]

What are a and d?
Can you express T1, T3 and T7 as functions of a and d? Do you know what "geometric pattern" means?

 

[TheRedDevil18]

Okay,
T1 = a
T3 = a+2d
T7 = a+6d

A geometric pattern is one with a constant ratio

 

[mfb]

A geometric pattern is one with a constant ratio

Don't just say it, express it as formula :).

 

[TheRedDevil18]

Tn = ar^n-1

 

[mfb]

Oh come on, you can try more than one step per post...

 

[vela]

S3 = T1+T3+T7

This isn't correct. The third partial sum, S3, is always T1+T2+T3.

The problem statement didn't say anything about S3. It just said T1, T3, and T7 are the first three terms of a geometric progression.

 

[TheRedDevil18]

This isn't correct. The third partial sum, S3, is always T1+T2+T3.

The problem statement didn't say anything about S3. It just said T1, T3, and T7 are the first three terms of a geometric progression.

So how would I bring the geometric pattern into all of this?

 

[vela]

That's what you're supposed to figure out.

You should use different letters here for the two different sequences. You've used T_n to denote the terms in the arithmetic sequence, and S_n to denote the partial sums formed from T_n. So to summarize what you have so far:
\begin{align*}
T_n &= a+(n-1)d \\
S_9 &= T_1 + T_2 + \cdots + T_9 = \frac{9}{2}(2a+8d)
\end{align*} You also said a geometric sequence has terms of the form $G_n = br^{n-1}$. I used G instead of T because you have two different sequences, and I used $b$ instead of $a$ because you already used the letter $a$ as one of the parameters for the arithmetic sequence.

In terms of the T's and G's, how would you express the sentence "The first, third and seventh terms of the (arithmetic) series form the first three terms of a geometric pattern"?

 

[TheRedDevil18]

(a)+(a+2d)+(a+6d) = (a)+(ar)+(ar^2)

 

[mfb]

Why do you want to add anything? Each term of your series should correspond to one term of the geometric series. This gives 3 equations without a sum, not one.
As vela pointed out, please do not use "a" for two different things. Use b instead for the geometric series. In addition, it could help if you follow vela's advice and use T_n, G_n first for those equations. You can plug in the known formulas for them afterwards.

 

[TheRedDevil18]

Ok so,
a = b
a+2d = br
a+6d = br^2

 

[vela]

Looks good. Keep going.

 

[Zimathster]

Since they are in geometric series. Then (a+7d)/(a+2d) = (a+2d)/(a)

 

[Zimathster]

From the first equation about the sum of the 9 terms u can write a equation in terms of a and d , then use system of equation to solve for a and d