Sequences and Series Problem

1. Apr 24, 2013

TheRedDevil18

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. The attempt at a solution

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

Completely stuck and need help

Last edited by a moderator: Apr 24, 2013
2. Apr 24, 2013

Staff: Mentor

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?

3. Apr 24, 2013

TheRedDevil18

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

A geometric pattern is one with a constant ratio

4. Apr 24, 2013

Staff: Mentor

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

5. Apr 24, 2013

TheRedDevil18

Tn = ar^n-1

6. Apr 24, 2013

Staff: Mentor

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

7. Apr 24, 2013

vela

Staff Emeritus
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.

8. Apr 24, 2013

TheRedDevil18

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

9. Apr 24, 2013

vela

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

You should use different letters here for the two different sequences. You've used Tn to denote the terms in the arithmetic sequence, and Sn to denote the partial sums formed from Tn. 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"?

10. Apr 25, 2013

TheRedDevil18

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

11. Apr 25, 2013

Staff: Mentor

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 Tn, Gn first for those equations. You can plug in the known formulas for them afterwards.

12. Apr 25, 2013

TheRedDevil18

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

13. Apr 25, 2013

vela

Staff Emeritus
Looks good. Keep going.

14. Apr 25, 2013

Zimathster

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

15. Apr 25, 2013

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