Aesity's question at Yahoo Answers regarding a geometric sequence

In summary, the seventh term of the geometric sequence is $\displaystyle \frac{1024 \cdot 2^{\frac{4}{5}}}{27}$ and the sum of the first five terms is $\displaystyle \frac{269}{3(2\cdot2^{\frac{4}{5}}-3)}$.
  • #1
MarkFL
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Here is the question:

Geometric sequence question?

The first term of a geometric sequence is 27 and the sixth term is 512/9.

1) Find the seventh term

I have problem with this working. Either the calculator is pulling a false one on me or IDK.
Since 512/9 = 27r^(6-1)
then (512/9) / 27 = r^5
r^5=512/243
r= (512/243) ^ 1/5
What I got for r was 0.421399177

To arrive at the seventh term; 27(0.421399177)^(7-1). I got 0.15119
This is no where close to the answer but apparently there doesn't appear to be any anomalies with my workings. 2) Find the sum of the first five terms of this sequence.

I need explanations behind the workings. Thanks!

Here is a link to the question:

Geometric sequence question? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.
 
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  • #2
Re: Aesity's question at Yahoo! Ansers regarding a geometric sequence

Hello Aesity,

We are given:

$\displaystyle a_1=27$

$\displaystyle a_6=\frac{512}{9}$

The $n$th term in the sequence is:

$\displaystyle a_n=27r^{n-1}$

and so we may determine $r$ by writing:

$\displaystyle a_6=\frac{512}{9}=27r^{6-1}$

$\displaystyle r^5=\frac{2^9}{3^5}$

$\displaystyle r=\frac{2^{\frac{9}{5}}}{3}=\frac{2\cdot2^{\frac{4}{5}}}{3}$

and so we have:

$\displaystyle a_n=27\left(\frac{2\cdot2^{\frac{4}{5}}}{3} \right)^{n-1}$

1.) Thus, we may now answer the first question:

$\displaystyle a_7=27\left(\frac{2\cdot2^{\frac{4}{5}}}{3} \right)^{6}=\frac{64\cdot2^{\frac{24}{5}}}{27}= \frac{1024 \cdot 2^{\frac{4}{5}}}{27}$

To find the sum of the first $n$ terms, consider:

$\displaystyle S_n=a_1+a_2+a_3+\cdots+a_n$

$\displaystyle S_n=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}$

Multiply through by $r$:

$\displaystyle rS_n=a_1r+a_1r^2+a_1r^3+\cdots+a_1r^{n}=S_n+a_1r^{n}-a_1$

$\displaystyle rS_n-S_n=a_1r^{n}-a_1$

$\displaystyle (r-1)S_n=a_1r^{n}-a_1$

$\displaystyle S_n=\frac{a_1r^{n}-a_1}{r-1}=a_1\cdot\frac{r^n-1}{r-1}$

2.) Now we may answer the second question:

$\displaystyle S_5=a_1\cdot\frac{r^5-1}{r-1}=27\cdot\frac{\left(\frac{2\cdot2^{\frac{4}{5}}}{3} \right)^5-1}{\frac{2\cdot2^{\frac{4}{5}}}{3}-1}=\frac{269}{3(2\cdot2^{\frac{4}{5}}-3)}$
 
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Related to Aesity's question at Yahoo Answers regarding a geometric sequence

1. What is a geometric sequence?

A geometric sequence is a type of numerical sequence where the ratio between any two consecutive terms is constant. This means that each term is multiplied by the same number to get the next term in the sequence.

2. How do you find the common ratio of a geometric sequence?

To find the common ratio of a geometric sequence, you divide any term by the previous term. For example, if the first term is 2 and the second term is 6, the common ratio would be 6/2 = 3.

3. What is the formula for finding the nth term of a geometric sequence?

The formula for finding the nth term of a geometric sequence is:
an = a1 * rn-1
Where an is the nth term, a1 is the first term, and r is the common ratio.

4. Can a geometric sequence have negative terms?

Yes, a geometric sequence can have negative terms. As long as the ratio between any two consecutive terms is constant, the sequence is still considered geometric.

5. How can geometric sequences be applied in real life?

Geometric sequences can be used to model growth or decay in various situations, such as population growth, compound interest, and radioactive decay. They can also be used in fields like physics and engineering to analyze patterns and make predictions.

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